#6379472

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×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(6,10)(11,12)(13,14)(15,23)(16,27)(17,21)(18,31)(19,25)(20,29)(22,30)(24,39)(26,34)(28,36)(32,35)(33,43)(37,41)(38,42)(40,44);(5,6,9,10)(7,11,8,12)(13,15,17,25)(14,19,21,23)(16,24,28,40)(18,26,30,42)(20,33,35,37)(22,34,31,38)(27,44,36,39)(29,41,32,43);(5,7,9,8)(6,11,10,12)(13,16,17,28)(14,20,21,35)(15,24,25,40)(18,32,30,29)(19,33,23,37)(22,36,31,27)(26,43,42,41)(34,39,38,44);(5,8,9,7)(6,11,10,12)(15,25)(19,23)(24,40)(26,42)(33,37)(34,38)(39,44)(41,43);(14,21)(15,25)(20,35)(22,31)(24,40)(26,42)(27,36)(41,43)

Polycyclic presentation:

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

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, 6286668);