#6536152

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2075 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)(3,8)(4,6)(5,7)(17,18,22,26)(19,23,27,31)(20,25,21,24)(28,32,29,30);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,23,26,31)(20,28,21,29)(24,30,25,32);(10,13)(14,16)(18,24)(20,21)(23,30)(25,26)(28,29)(31,32);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,27)(23,31)(28,29)(30,32);(11,15)(14,16)(17,20,22,21)(18,25,26,24)(19,28,27,29)(23,32,31,30)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281564, #72282791, #72282805, …
maximal normal subgroup of	#72281564, #72282791, #72282805, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443348);