#6442529

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4674 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)(11,20)(12,17)(13,18)(14,27)(15,19)(16,22)(21,34)(23,30)(24,32)(25,33)(26,37)(28,31)(29,35)(36,39)(38,40);(5,6)(7,8)(9,11,15,23)(10,16,19,31)(12,21,25,36)(13,22,14,28)(17,29,33,40)(18,30,27,20)(24,35,26,38)(32,39,37,34);(1,2,3,4)(6,8)(9,12)(10,17)(11,21)(13,24)(14,26)(15,25)(16,29)(18,32)(19,33)(20,34)(22,35)(23,36)(27,37)(28,38)(30,39)(31,40);(6,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,25,26)(16,30,31,20)(17,32,33,37)(21,35,36,38)(29,39,40,34);(12,25)(17,33)(21,36)(24,26)(29,40)(32,37)(34,39)(35,38)

Polycyclic presentation:

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

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, 6349725);