#6458451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1558 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)(12,16)(13,15)(17,18,22,26,24,19,23,21)(20,25,31,28,30,27,32,29);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,28)(21,23)(22,26)(25,31)(27,32)(29,30);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,22,31,24,30,23,32)(18,25,26,28,19,27,21,29);(1,4,5,3)(2,8,6,7)(17,21,22,18,24,26,23,19)(20,29,31,25,30,28,32,27);(20,30)(25,27)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=fg,d^2=f,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^c=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, 6365647);