#6584967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#35941	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,17,22)(15,21,25,19)(16,20,18,23)(24,26,28,27);(6,10)(11,12)(13,14)(15,24)(16,23)(17,22)(18,20)(19,26)(21,27)(25,28);(1,2,3,4)(13,15,17,25)(14,19,22,21)(16,26,18,27)(20,24,23,28);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,25)(17,18)(19,21)(22,23);(5,8,9,7)(6,12,10,11)(13,15,17,25)(14,21,22,19)(16,26,18,27)(20,28,23,24)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492163);