#6814568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	C₂×#3203	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721764);