#146701

order	512
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10761
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

985 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 188 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=di,f^b=fh,e^c=ei,f^c=fhi,f^d=fi,f^e=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, 53897);