#6458603

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1773 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)(9,10,13,15)(11,14,12,16)(17,18,22,27,23,28,21,26)(19,29,30,24,31,32,20,25);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,22)(21,30)(23,31)(25,28)(26,32)(27,29);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,18,22,27,23,28,21,26)(19,29,30,24,31,32,20,25);(1,4,5,3)(2,8,6,7)(17,20,21,31,23,30,22,19)(18,25,26,32,28,24,27,29);(17,21,23,22)(18,26,28,27)(19,30,31,20)(24,32,25,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fgi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,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, 6365799);