#7001231

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1891 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257286, #72257525, #72264103
maximal normal subgroup of	#72257286, #72257525, #72264103
maximal quotient of	#72265670

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6908427);