#7183395

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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fhi,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7090591);