#7138683

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	#45271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(17,18,23,24,22,26,19,25)(20,29,27,31,30,28,32,21);(1,2)(3,6)(4,5)(7,8)(10,11)(12,13)(17,19,22,23)(18,24,26,25)(20,27,30,32)(21,28,31,29);(9,11,14,10)(12,15,13,16)(17,20)(18,21)(19,27)(22,30)(23,32)(24,28)(25,29)(26,31);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,21)(18,20)(19,28)(22,31)(23,29)(24,27)(25,32)(26,30);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,19,22,23)(18,25,26,24)(20,27,30,32)(21,29,31,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=h,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, 7045879);