#6584908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1965 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)(13,14,18,19)(15,22,25,20)(16,17,21,27)(23,26,28,24);(6,10)(11,12)(13,15)(14,20)(16,26)(17,23)(18,25)(19,22)(21,24)(27,28);(1,2,3,4)(13,16)(14,17)(15,23)(18,21)(19,27)(20,24)(22,26)(25,28);(5,7,9,8)(6,11,10,12)(14,19)(16,27)(17,21)(20,22)(23,24)(26,28);(5,8,9,7)(6,12,10,11)(13,17,18,27)(14,21,19,16)(15,24,25,26)(20,28,22,23)

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,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492104);