#6464899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6372095);