#6437785

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6344981);