#7144115

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42805	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,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, 7051311);