#6465564

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3226 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6372760);