#7307957

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300884
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	#23857	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 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)(4,7)(5,8)(17,18)(19,27)(20,25)(21,26)(22,24)(23,28)(29,32)(30,31);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,28)(18,23,26,27)(20,29,22,30)(24,32,25,31);(9,11)(10,13)(12,16)(14,15)(18,24)(19,28)(20,22)(23,31)(25,26)(27,32);(1,4,6,5)(2,7,3,8)(10,14)(11,16)(17,20,21,22)(18,25,26,24)(19,29,28,30)(23,31,27,32);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(18,26)(19,28)(24,25)(29,30)

Polycyclic presentation:

2^9|b^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7215153);