#7283624

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
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	#42889	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7190820);