#388351

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295547);