#6383847

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	C₂×#3285	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2613 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)(3,4)(5,6,8,7)(9,10,16,22)(11,23,28,34)(12,18,13,19)(14,21,15,20)(17,26,35,27)(24,33,30,40)(25,32,39,37)(29,38,31,36);(9,11,16,28)(10,17,22,35)(12,24,13,30)(14,27,15,26)(18,32,19,37)(20,23,21,34)(25,29,39,31)(33,36,40,38);(2,4)(9,12,16,13)(10,18,22,19)(11,24,28,30)(14,31,15,29)(17,32,35,37)(20,38,21,36)(23,33,34,40)(25,26,39,27);(9,13,16,12)(10,19,22,18)(11,25,28,39)(14,31,15,29)(17,33,35,40)(20,38,21,36)(23,32,34,37)(24,26,30,27);(5,7,8,6)(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,29,13,31)(17,34,35,23)(18,36,19,38)(24,25,30,39)(32,33,37,40)

Polycyclic presentation:

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

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, 6291043);