#389089

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1302 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)(9,10,13,18,14,19,28,16)(11,15,20,30,23,32,22,31)(12,24,25,34,27,33,26,17)(21,36,37,39,38,40,35,29);(1,3,4,2)(9,11)(10,15)(12,21)(13,20)(14,23)(16,31)(17,29)(18,30)(19,32)(22,28)(24,36)(25,37)(26,35)(27,38)(33,40)(34,39);(5,7,8,6)(10,16)(11,20)(13,28)(17,24)(18,19)(21,35)(22,23)(25,26)(30,31)(33,34)(36,40)(37,38);(2,3)(9,12,14,27)(10,17,19,34)(11,21,23,38)(13,26,28,25)(15,29,32,39)(16,24,18,33)(20,35,22,37)(30,40,31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=i,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, 296285);