#6915513

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

1528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,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, #72308911, #72309046
maximal normal subgroup of	#72308682, #72308911, #72309046

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6822709);