#6891442

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(9,10)(11,16)(12,21)(13,17)(14,18)(15,20)(19,23)(22,24)(25,26)(27,34)(28,36)(29,33)(30,31)(32,38)(35,39)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,18)(17,23)(21,24)(25,27,29,30)(26,31,33,34)(28,37,38,35)(32,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,19)(14,20)(15,18)(17,23)(21,24)(26,31)(27,30)(32,39)(33,34)(35,37)(36,40);(3,8)(5,7)(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,23,16,24)(25,28,29,38)(26,32,33,36)(27,35,30,37)(31,39,34,40);(2,6)(5,7)(9,13,14,12)(10,17,18,21)(11,19,20,22)(15,24,16,23)(28,38)(32,36)(35,37)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308725, #72308985
maximal normal subgroup of	#72308725, #72308985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6798638);