#6381873

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2620 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289069);