#7138685

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1414 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)(6,8)(9,10)(11,14)(12,15)(13,16)(17,18,24,27,23,20,22,28)(19,29,21,26,30,25,31,32);(1,2)(3,6)(4,5)(7,8)(10,11)(12,13)(17,19,23,30)(18,25,20,29)(21,22,31,24)(26,27,32,28);(9,11,14,10)(12,15,13,16)(17,20,23,18)(19,25,30,29)(21,26,31,32)(22,28,24,27);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,21)(18,26)(19,22)(20,32)(23,31)(24,30)(25,27)(28,29);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,22,23,24)(18,27,20,28)(19,21,30,31)(25,26,29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7045881);