#7129140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fh,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7036336);