#7513514

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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,22,26,30)(20,24,28,32)(25,29,27,31);(1,3)(2,4)(5,7)(6,8)(10,13)(11,15)(17,19,21,26)(18,22,23,30)(20,25,28,27)(24,29,32,31);(10,13)(14,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16)(18,23)(22,30)(24,32)(29,31);(1,5,6,4)(2,3,7,8)(19,26)(22,30)(25,27)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,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, 7420710);