#6443133

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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

4964 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6350329);