#6683693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2422 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)(13,14)(15,21)(16,27)(17,22)(18,23)(19,25)(20,24)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,23)(17,20)(18,27)(19,26)(22,24)(25,28);(2,4)(13,16,19,22,20,28,18,21)(14,17,23,27,24,15,25,26);(5,7,8,11)(6,10,9,12)(13,17,20,15)(14,22,24,21)(16,23,28,25)(18,26,19,27);(2,4)(6,9)(10,12)(13,18,20,19)(14,23,24,25)(15,26,17,27)(16,21,28,22)

Polycyclic presentation:

2^9|c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6590889);