#7312354

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#53060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 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)(17,18,22,26,23,25,31,32)(19,20,27,28,21,24,29,30);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,21)(18,20,25,24)(22,27,31,29)(26,28,32,30);(9,11)(10,13)(12,16)(14,15)(17,20,23,24)(18,19,25,21)(22,30,31,28)(26,29,32,27);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,30,32,28);(2,5)(6,8)(10,14)(13,15)(18,25)(19,21)(26,32)(27,29)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

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, 7219550);