#6437871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3487	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4704 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345067);