#6914361

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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

1604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,e^2=f,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,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308682, #72308965, #72309061
maximal normal subgroup of	#72308682, #72308965, #72309061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6821557);