#6808371

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=f,e^2=h,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

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, 6715567);