#6444272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	C₂×#3054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351468);