#6753055

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
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	#36096	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2102 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)(3,7)(4,6)(5,8)(17,18)(19,27)(20,26)(21,25)(22,24)(23,29)(28,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,27)(26,32);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(18,24)(20,22)(23,31)(25,26)(27,32)(28,30);(2,6)(5,7)(18,25)(23,27)(24,26)(31,32);(10,14)(13,15)(17,20,21,22)(18,26,25,24)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256949, #72256986, #72256993
maximal normal subgroup of	#72256949, #72256986, #72256993

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6660251);