#6437775

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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 ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4740 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)(9,10,16,30,15,21,14,20)(11,22,29,17,26,36,25,33)(12,18,13,19,27,34,28,35)(23,32,39,31,38,40,24,37);(5,6)(7,8)(9,11)(10,17)(12,23)(13,24)(14,29)(15,26)(16,25)(18,31)(19,32)(20,36)(21,33)(22,30)(27,38)(28,39)(34,37)(35,40);(2,4)(6,8)(9,12,15,27)(10,18,21,34)(11,23,26,38)(13,14,28,16)(17,31,33,37)(19,20,35,30)(22,32,36,40)(24,29,39,25);(6,8)(9,13)(10,19)(11,24)(12,14)(15,28)(16,27)(17,32)(18,20)(21,35)(22,37)(23,29)(25,38)(26,39)(30,34)(31,36)(33,40);(12,27)(13,28)(18,34)(19,35)(23,38)(24,39)(31,37)(32,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=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, 6344971);