#6381888

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,g^c=gi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289084);