#495767

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₄×B(4,𝔽₂)
prime radical	#33564	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1005 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1005 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,14,9,15,7,31)(3,16,20,45,21,43,19,33)(4,13,23,11,6,28,30,39)(5,12,26,41,27,42,25,55)(10,32,36,29,37,50,35,17)(18,44,48,63,49,62,47,57)(22,40,51,38,24,52,54,61)(34,56,59,53,60,64,58,46);(1,3)(2,10)(4,17)(5,18)(6,29)(7,19)(8,20)(9,21)(11,33)(12,34)(13,43)(14,36)(15,37)(16,28)(22,46)(23,32)(24,53)(25,47)(26,48)(27,49)(30,50)(31,35)(38,57)(39,45)(40,62)(41,59)(42,60)(44,52)(51,56)(54,64)(55,58)(61,63);(1,4)(2,11)(3,17)(5,22)(6,9)(7,30)(8,23)(10,33)(12,38)(13,31)(14,28)(15,39)(16,36)(18,46)(19,50)(20,32)(21,29)(24,27)(25,54)(26,51)(34,57)(35,43)(37,45)(40,55)(41,52)(42,61)(44,59)(47,64)(48,56)(49,53)(58,62)(60,63);(1,5,8,26,9,27,7,25)(2,12,14,41,15,42,31,55)(3,18,20,48,21,49,19,47)(4,22,23,51,6,24,30,54)(10,34,36,59,37,60,35,58)(11,38,28,52,39,61,13,40)(16,44,45,63,43,62,33,57)(17,46,32,56,29,53,50,64)

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317420
maximal normal subgroup of	#72317420

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402963);