#6466440

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1435 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,32);(2,5)(6,8)(17,19,23,20)(18,25,28,30)(21,24,31,22)(26,27,32,29);(9,11,12,16)(10,13,14,15)(17,20,24,31,23,19,22,21)(18,25,29,32,28,30,27,26);(17,21,24,20,23,31,22,19)(18,26,29,25,28,32,27,30);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,27,28,29)(19,21,20,31)(25,32,30,26)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,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, 6373636);