#388326

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1320 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)(11,21)(12,26)(13,19)(14,17)(15,20)(16,24)(18,30)(22,33)(23,38)(25,32)(27,40)(28,34)(29,36)(31,37)(35,39);(1,2)(3,4)(9,11)(10,16)(12,27)(13,24)(14,25)(15,22)(17,33)(18,35)(19,21)(20,32)(23,28)(26,38)(29,39)(30,37)(31,36)(34,40);(5,7,8,6)(10,17)(11,22)(14,15)(18,34)(19,20)(23,37)(24,25)(26,36)(27,39)(29,30)(32,33)(38,40);(2,4)(9,12)(10,18)(11,23)(13,28)(14,30)(15,29)(16,31)(17,34)(19,36)(20,26)(21,35)(22,37)(24,27)(25,39)(32,40)(33,38)

Polycyclic presentation:

2^9|a^2=h,c^2=ehi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295522);