#6442523

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437415
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×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4624 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)(9,10,15,20)(11,21,24,34)(12,17,25,31)(13,19,14,18)(16,28,30,23)(22,35,36,40)(26,33,27,32)(29,38,39,37);(5,6)(7,8)(9,11)(10,16)(12,22)(13,23)(14,28)(15,24)(17,29)(18,21)(19,34)(20,30)(25,36)(26,37)(27,38)(31,39)(32,35)(33,40);(1,2,3,4)(6,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,25)(16,29)(18,32)(19,33)(20,31)(21,35)(23,37)(24,36)(28,38)(30,39)(34,40);(6,8)(12,25)(17,31)(22,36)(26,27)(29,39)(32,33)(35,40)(37,38);(9,13,15,14)(10,18,20,19)(11,23,24,28)(12,26,25,27)(16,21,30,34)(17,32,31,33)(22,37,36,38)(29,35,39,40)

Polycyclic presentation:

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

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, 6349719);