#6438047

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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

4700 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)(9,10,13,16,15,20,14,19)(11,21,28,17,24,34,23,31)(12,18,27,29,26,33,25,32)(22,35,36,30,38,39,37,40);(2,4)(5,6)(7,8)(10,16)(13,14)(17,21)(18,29)(19,20)(23,28)(25,27)(30,35)(31,34)(32,33)(36,37)(39,40);(6,8)(9,11,15,24)(10,17,20,31)(12,22,26,38)(13,23,14,28)(16,21,19,34)(18,30,33,40)(25,36,27,37)(29,35,32,39);(6,8)(9,12)(10,18)(11,22)(13,27)(14,25)(15,26)(16,29)(17,30)(19,32)(20,33)(21,35)(23,37)(24,38)(28,36)(31,40)(34,39);(9,13,15,14)(10,16,20,19)(11,23,24,28)(12,25,26,27)(17,21,31,34)(18,32,33,29)(22,36,38,37)(30,39,40,35)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 6345243);