#6682990

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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

2210 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)(15,22)(16,24)(17,18)(19,20)(21,25)(23,26)(27,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,18,19)(14,20,17,22)(16,25,27,23)(21,24,26,28);(2,4)(13,16,19,25,18,27,15,23)(14,21,22,24,17,26,20,28);(6,10)(9,11)(13,17,18,14)(15,22,19,20)(16,26,27,21)(23,24,25,28);(2,4)(7,12)(9,11)(13,15,18,19)(14,22,17,20)(16,23,27,25)(21,24,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283358, #72283671, #72284414
maximal normal subgroup of	#72283358, #72283671, #72284414

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590186);