#6558467

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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

2036 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)(19,25)(20,26)(22,28)(24,29)(27,31)(30,32);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(10,13)(14,16)(17,19,26,31,21,27,20,25)(18,22,29,32,23,30,24,28);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(18,23)(22,30)(24,29)(28,32);(11,15)(14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256894, #72256928, #72256963
maximal normal subgroup of	#72256894, #72256928, #72256963
maximal quotient of	#72264741

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465663);