#6810554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25736	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 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)(13,14,18,20)(15,24,25,36)(16,28,30,22)(17,23,19,32)(21,33,34,27)(26,41,40,39)(29,38,31,37)(35,43,44,42);(2,3)(5,7,8,12)(6,9,10,11)(14,20)(15,25)(22,28)(23,32)(26,40)(27,33)(37,38)(42,43);(1,2,4,3)(7,12)(9,11)(13,15)(14,21)(16,26)(17,27)(18,25)(19,33)(20,34)(22,35)(23,36)(24,32)(28,44)(29,42)(30,40)(31,43)(37,39)(38,41);(6,10)(9,11)(13,16,18,30)(14,22,20,28)(15,26,25,40)(17,31,19,29)(21,35,34,44)(23,38,32,37)(24,39,36,41)(27,43,33,42);(1,3,4,2)(13,17,18,19)(14,23,20,32)(15,27,25,33)(16,29,30,31)(21,36,34,24)(22,37,28,38)(26,42,40,43)(35,39,44,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fh,d^2=i,e^2=fhi,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, 6717750);