#6624810

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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
prime radical	#28584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(18,22)(19,25)(20,26)(23,30)(24,29)(27,31);(2,6)(7,8)(10,14)(11,12)(17,18,20,22,21,24,26,29)(19,23,25,28,27,30,31,32);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19,26,31,21,27,20,25)(18,23,29,32,24,30,22,28);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(18,24)(22,29)(23,30)(28,32);(9,12,13,11)(10,16,14,15)(19,27)(23,30)(25,31)(28,32)

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313310, #72313687, #72313813
maximal normal subgroup of	#72313310, #72313687, #72313813

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6532006);