#129593

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8316
prime radical	#3529	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

601 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=egi,f^b=fgi,d^c=dg,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36789);