#6463975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6371171);