#6468007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=hi,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 6375203);