#6546276

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#24388	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6453472);