#6383931

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	C₂×#3283	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2654 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)(13,26)(14,17)(15,22)(18,27)(20,33)(21,29)(23,32)(24,37)(25,30)(28,35)(31,38)(34,39)(36,40);(9,11)(10,15)(12,20)(13,21)(14,23)(16,28)(17,29)(18,32)(19,26)(22,27)(24,34)(25,36)(30,39)(31,40)(33,37)(35,38);(2,4)(9,12,14,25)(10,16,17,30)(11,20,23,36)(13,24,27,38)(15,28,29,39)(18,31,26,37)(19,33,32,40)(21,34,22,35);(10,17)(11,21)(13,27)(15,19)(16,30)(20,34)(22,23)(24,38)(28,33)(29,32)(35,36)(39,40);(5,7,8,6)(10,17)(15,29)(16,30)(18,26)(19,32)(28,39)(31,37)(33,40)

Polycyclic presentation:

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

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, 6291127);