#6682864

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2286 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590060);