#6464898

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

1449 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)(17,18,24,21,23,28,22,27)(19,29,26,25,32,20,31,30);(2,5)(6,8)(17,19)(18,25)(20,27)(21,29)(22,26)(23,32)(24,31)(28,30);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,25,23,29,24,30)(18,26,27,19,28,31,21,32);(17,21,22,18,23,27,24,28)(19,30,31,20,32,25,26,29);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,21,28,27)(19,26,32,31)(20,25,29,30)

Polycyclic presentation:

2^9|a^2=fi,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, 6372094);