#6810578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2015 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)(13,14,19,24)(15,25,28,38)(16,21,29,35)(17,23,18,22)(20,32,34,27)(26,39,40,44)(30,37,31,36)(33,41,43,42);(2,3)(5,7,8,12)(6,9,10,11)(16,29)(21,35)(26,40)(30,31)(33,43)(36,37)(39,44)(41,42);(1,2,4,3)(7,12)(9,11)(13,15)(14,20)(16,26)(17,27)(18,32)(19,28)(21,33)(22,25)(23,38)(24,34)(29,40)(30,41)(31,42)(35,43)(36,44)(37,39);(6,10)(9,11)(13,16,19,29)(14,21,24,35)(15,26,28,40)(17,31,18,30)(20,33,34,43)(22,37,23,36)(25,39,38,44)(27,42,32,41);(1,3,4,2)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,30,29,31)(20,25,34,38)(21,36,35,37)(26,41,40,42)(33,44,43,39)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717774);