#6458579

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=fg,d^2=f,e^2=i,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, 6365775);