#6463579

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 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)(17,18,23,21)(19,22,26,24)(20,28,30,27)(25,32,31,29);(2,5)(6,8)(11,12)(14,16)(17,19)(18,24)(20,27)(21,22)(23,26)(25,29)(28,30)(31,32);(9,11,13,12)(10,14,15,16)(17,20,22,32,23,30,24,29)(18,25,19,27,21,31,26,28);(9,12,13,11)(10,14,15,16)(17,21,23,18)(19,22,26,24)(20,25,30,31)(27,29,28,32);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,26,21,19)(20,29,30,32)(25,27,31,28)

Polycyclic presentation:

2^9|a^2=i,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=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370775);