#6383855

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×#3277	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)(9,10,16,21)(11,22,27,34)(12,18,13,30)(14,20,15,19)(17,25,35,26)(23,33,28,40)(24,36,39,32)(29,38,31,37);(9,11,16,27)(10,17,21,35)(12,28,13,23)(14,26,15,25)(18,36,30,32)(19,22,20,34)(24,29,39,31)(33,38,40,37);(2,4)(9,12,16,13)(10,18,21,30)(11,23,27,28)(14,31,15,29)(17,32,35,36)(19,38,20,37)(22,33,34,40)(24,26,39,25);(9,13,16,12)(10,18,21,30)(11,24,27,39)(14,31,15,29)(17,33,35,40)(19,37,20,38)(22,32,34,36)(23,26,28,25);(5,7,8,6)(9,14,16,15)(10,19,21,20)(11,25,27,26)(12,29,13,31)(17,34,35,22)(18,37,30,38)(23,39,28,24)(32,33,36,40)

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,d^a=di,e^a=ei,f^a=fi,c^b=ci,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, 6291051);