#6683071

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2060 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590267);