#6810355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1969 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)(13,14,18,23)(15,25,28,20)(16,29,32,40)(17,24,19,22)(21,34,38,31)(26,41,30,44)(27,36,33,39)(35,43,37,42);(2,3)(5,7,8,12)(6,9,10,11)(13,15,18,28)(14,20,23,25)(16,30,32,26)(17,33,19,27)(21,37,38,35)(22,39,24,36)(29,41,40,44)(31,42,34,43);(1,2,4,3)(7,12)(9,11)(13,16,18,32)(14,21,23,38)(15,26,28,30)(17,31,19,34)(20,35,25,37)(22,29,24,40)(27,42,33,43)(36,41,39,44);(6,10)(9,11)(13,17,18,19)(14,22,23,24)(15,27,28,33)(16,31,32,34)(20,36,25,39)(21,29,38,40)(26,42,30,43)(35,41,37,44);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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,d^b=di,e^b=efh,g^c=gi

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, 6717551);