#6444279

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2840 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)(9,10,15,19)(11,21,13,16)(12,25,28,35)(14,18,30,20)(17,27,33,29)(22,37,26,40)(23,34,24,32)(31,38,36,39);(1,2)(3,4)(6,7)(9,11,15,13)(10,16,19,21)(12,22,28,26)(14,24,30,23)(17,31,33,36)(18,34,20,32)(25,40,35,37)(27,39,29,38);(2,4)(5,6,8,7)(9,12,14,29,15,28,30,27)(10,17,20,25,19,33,18,35)(11,22,24,38,13,26,23,39)(16,31,32,40,21,36,34,37);(2,4)(5,7,8,6)(9,13,15,11)(10,16,19,21)(12,26,28,22)(14,23,30,24)(17,31,33,36)(18,34,20,32)(25,40,35,37)(27,38,29,39);(9,14,15,30)(10,18,19,20)(11,23,13,24)(12,27,28,29)(16,32,21,34)(17,25,33,35)(22,38,26,39)(31,37,36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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=dh,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351475);