#6443127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5080 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)(11,21)(12,18)(13,19)(14,29)(15,20)(16,24)(17,27)(22,36)(23,32)(25,37)(26,34)(28,33)(30,40)(31,38)(35,39);(5,6)(7,8)(9,11,14,24,15,25,13,23)(10,16,19,21,20,32,29,37)(12,22,26,38,27,39,28,40)(17,30,34,35,18,31,33,36);(1,2,3,4)(6,8)(9,12)(10,17)(11,22)(13,28)(14,26)(15,27)(16,30)(18,20)(19,34)(21,35)(23,40)(24,38)(25,39)(29,33)(31,32)(36,37);(6,8)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,31)(17,20)(19,33)(21,36)(23,40)(24,38)(25,39)(29,34)(30,32)(35,37);(9,13,15,14)(10,19,20,29)(11,23,25,24)(12,26,27,28)(16,21,32,37)(17,33,18,34)(22,38,39,40)(30,36,31,35)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6350323);