#6438042

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×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4636 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)(9,10,14,20,16,22,15,21)(11,23,25,36,27,35,26,17)(12,28,30,18,24,37,29,38)(13,19,32,40,33,34,31,39);(2,4)(5,6)(7,8)(9,11,16,27)(10,17,22,36)(12,13,24,33)(14,26,15,25)(18,34,38,19)(20,35,21,23)(28,39,37,40)(29,32,30,31);(6,8)(9,12,16,24)(10,18,22,38)(11,13,27,33)(14,29,15,30)(17,34,36,19)(20,28,21,37)(23,40,35,39)(25,31,26,32);(6,8)(9,13)(10,19)(11,24)(12,27)(14,32)(15,31)(16,33)(17,18)(20,40)(21,39)(22,34)(23,37)(25,29)(26,30)(28,35)(36,38);(9,14,16,15)(10,20,22,21)(11,25,27,26)(12,29,24,30)(13,31,33,32)(17,23,36,35)(18,28,38,37)(19,39,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 6345238);