#6463381

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#28149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(19,25)(20,27)(22,29)(24,30)(26,31)(28,32);(2,5)(6,8)(11,12)(14,16)(17,18)(19,26)(20,24)(21,23)(22,28)(25,31)(27,30)(29,32);(9,11,13,12)(10,14,15,16)(17,19,27,32,21,28,20,25)(18,22,30,31,23,26,24,29);(9,12,13,11)(10,14,15,16);(1,3,4,7)(2,6,5,8)(18,23)(19,28)(24,30)(25,32)

Polycyclic presentation:

2^9|c^2=gi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370577);