#7233734

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200082
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	#46333	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1932 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)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,32)(30,31);(2,5)(3,7)(10,14)(11,12)(18,22)(23,31)(24,29)(25,32);(9,11,13,12)(10,15,14,16)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,3,4,7)(2,6,5,8)(10,14)(15,16);(2,5)(6,8)(9,12,13,11)(10,15,14,16)

Polycyclic presentation:

2^9|c^2=hi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7140930);