#6383987

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×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2610 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)(11,19)(12,16)(13,27)(14,17)(15,23)(18,28)(20,35)(21,30)(22,34)(24,38)(25,39)(26,32)(29,37)(31,36)(33,40);(9,11,14,22)(10,15,17,30)(12,24,26,20)(13,21,28,23)(16,31,32,29)(18,34,27,19)(25,37,40,36)(33,35,39,38);(2,4)(9,12,14,26)(10,16,17,32)(11,20,22,24)(13,25,28,40)(15,29,30,31)(18,33,27,39)(19,35,34,38)(21,36,23,37);(10,17)(11,21)(13,28)(15,19)(16,32)(20,36)(22,23)(24,37)(25,40)(29,35)(30,34)(31,38);(5,7,8,6)(10,17)(11,22)(16,32)(18,27)(20,24)(21,23)(33,39)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^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,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, 6291183);