#6444252

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(5,6,8,7)(9,10,14,18)(11,15,20,28)(12,22,23,37)(13,25,26,17)(16,30,31,24)(19,32,34,40)(21,36,35,29)(27,38,39,33);(1,2)(3,4)(6,7)(9,11,14,20)(10,15,18,28)(12,19,23,34)(13,21,26,35)(16,27,31,39)(17,29,25,36)(22,32,37,40)(24,33,30,38);(2,4)(5,6,8,7)(9,12,13,24,14,23,26,30)(10,16,17,22,18,31,25,37)(11,19,21,33,20,34,35,38)(15,27,29,32,28,39,36,40);(2,4)(5,7,8,6);(11,20)(12,23)(15,28)(16,31)(21,35)(22,37)(24,30)(29,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=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, 6351448);