#7506962

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500661
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	#34715	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1633 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)(3,4)(5,7)(6,8)(9,10,12,14)(11,13,15,16)(18,23)(19,27)(25,26)(29,30);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,18,21,23)(19,28,27,32)(20,26,22,25)(24,30,31,29);(10,14)(13,16)(17,19,21,27)(18,24,23,31)(20,29,22,30)(25,28,26,32);(2,7)(3,8)(10,14)(13,16)(17,20,21,22)(18,25,23,26)(19,29,27,30)(24,28,31,32);(1,4,5,6)(2,3,7,8)(17,20,21,22)(18,26,23,25)(19,30,27,29)(24,28,31,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7414158);