#6618887

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#24128	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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)(3,6)(4,5)(7,8)(18,23)(19,26)(20,22)(24,31)(27,29)(30,32);(2,5)(6,8)(10,14)(12,15)(17,18,21,23)(19,24,27,30)(20,28,22,25)(26,32,29,31);(9,10)(11,12)(13,14)(15,16)(17,19)(18,24)(20,26)(21,27)(22,29)(23,30)(25,31)(28,32);(1,3,4,7)(2,6,5,8)(9,11,13,16)(10,15,14,12);(9,12,13,15)(10,11,14,16)(17,20,21,22)(18,25,23,28)(19,26,27,29)(24,31,30,32)

Polycyclic presentation:

2^9|b^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257398, #72257541, #72257782
maximal normal subgroup of	#72257398, #72257541, #72257782
maximal quotient of	#72308775

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6526083);