#389137

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

1180 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)(9,10,14,17,15,20,13,19)(11,16,23,31,24,32,21,30)(12,25,27,18,28,33,26,34)(22,36,35,29,38,39,37,40);(1,2)(3,4)(9,11)(10,16)(12,22)(13,21)(14,23)(15,24)(17,31)(18,29)(19,30)(20,32)(25,36)(26,37)(27,35)(28,38)(33,39)(34,40);(5,7,8,6)(10,17)(11,21)(13,14)(18,25)(19,20)(22,35)(23,24)(26,27)(30,31)(33,34)(36,39)(37,38);(2,4)(9,12)(10,18)(11,22)(13,27)(14,26)(15,28)(16,29)(17,25)(19,33)(20,34)(21,35)(23,37)(24,38)(30,39)(31,36)(32,40)

Polycyclic presentation:

2^9|a^2=h,c^2=efh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 296333);