#6379170

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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	C₂²×#578	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4601 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)(5,6,7,8)(10,15)(13,25)(16,28)(17,29)(18,26)(21,36)(22,32)(23,27)(30,39)(31,37)(33,40)(35,38);(9,10,14,18)(11,16,20,31)(12,22,24,29)(13,26,25,15)(17,27,32,23)(19,33,34,39)(21,37,36,28)(30,38,40,35);(2,4)(9,11)(10,16)(12,19)(13,21)(14,20)(15,28)(17,30)(18,31)(22,33)(23,35)(24,34)(25,36)(26,37)(27,38)(29,39)(32,40);(9,12,14,24)(10,17,18,32)(11,19,20,34)(13,27,25,23)(15,29,26,22)(16,30,31,40)(21,38,36,35)(28,39,37,33);(10,18)(11,20)(15,26)(17,32)(19,34)(21,36)(22,29)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,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, 6286366);