#6459321

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,24,27,23,26,22,20)(19,28,29,25,30,31,21,32);(2,6)(7,8)(17,19,23,30)(18,25,26,32)(20,31,27,28)(21,22,29,24);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,24,18,23,27,22,26)(19,25,29,31,30,32,21,28);(1,4,5,3)(2,8,6,7)(17,21,24,19,23,29,22,30)(18,25,27,31,26,32,20,28);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,20,26,27)(19,29,30,21)(25,31,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=fi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=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, 6366517);