#7145446

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	#44980	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18)(19,25)(20,23)(21,30)(22,28)(24,29)(26,31)(27,32);(2,5)(3,7)(10,13)(11,15)(17,18,20,23)(19,26,27,22)(21,24,29,30)(25,28,32,31);(9,10)(11,14)(12,13)(15,16)(17,19,20,27)(18,22,23,26)(21,31,29,28)(24,32,30,25);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,18)(19,26)(20,23)(21,24)(22,27)(25,31)(28,32)(29,30);(9,11,12,15)(10,14,13,16)(18,23)(22,26)(24,30)(25,32)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=hi,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=dh,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, 7052642);