#6810181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,11)(8,10)(9,12)(13,14,17,19)(15,23,24,36)(16,27,28,21)(18,31,30,22)(20,32,33,26)(25,39,38,37)(29,35,43,42)(34,40,41,44);(2,3)(5,7)(6,9)(8,12)(10,11)(14,19)(15,24)(21,27)(22,31)(25,38)(26,32)(35,42)(40,44);(1,2,4,3)(7,12)(9,11)(13,15,17,24)(14,20,19,33)(16,25,28,38)(18,32,30,26)(21,34,27,41)(22,36,31,23)(29,44,43,40)(35,39,42,37);(6,10)(9,11)(13,16,17,28)(14,21,19,27)(15,25,24,38)(18,29,30,43)(20,34,33,41)(22,35,31,42)(23,37,36,39)(26,40,32,44);(1,3,4,2)(16,28)(21,27)(25,38)(29,43)(34,41)(35,42)(37,39)(40,44)

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717377);