#7138807

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1483 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)(17,18,23,27,22,24,21,26)(19,28,29,30,25,31,32,20);(1,2)(3,6)(4,5)(7,8)(10,11)(12,13)(18,24)(20,30)(26,27)(28,31);(9,11,14,10)(12,15,13,16)(17,19,22,25)(18,20,24,30)(21,29,23,32)(26,28,27,31);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,20)(18,25)(19,24)(21,31)(22,30)(23,28)(26,29)(27,32);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,21,22,23)(18,26,24,27)(19,29,25,32)(20,28,30,31)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,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, 7046003);