#6468348

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1486 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)(9,10,13,16)(11,14,12,15)(17,18,24,28)(19,20,31,32)(21,26,25,30)(22,27,23,29);(11,12)(14,15)(17,19,23,26,24,31,22,30)(18,25,27,20,28,21,29,32);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20,24,32)(18,19,28,31)(21,22,25,23)(26,29,30,27);(9,12,13,11)(10,15,16,14)(17,21)(18,26)(19,29)(20,22)(23,32)(24,25)(27,31)(28,30);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,27,28,29)(19,30,31,26)(20,21,32,25)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375544);