#361138

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1282 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)(3,7)(4,5)(6,8)(9,10)(11,16,22,32)(12,24,13,17)(14,18)(15,19)(20,38)(21,31)(23,33,29,36)(25,40)(26,37,27,34)(28,35)(30,39);(2,5)(6,7)(9,11)(10,16)(12,25)(13,21)(14,22)(15,29)(17,31)(18,32)(19,36)(20,26)(23,28)(24,40)(27,39)(30,34)(33,35)(37,38);(9,12,14,13)(10,17,18,24)(11,20,22,39)(15,27,28,26)(16,30,32,38)(19,37,35,34)(21,29,25,23)(31,33,40,36);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,17,18,24)(11,21,22,25)(15,27,28,26)(16,31,32,40)(19,34,35,37)(20,29,39,23)(30,33,38,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,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, 268334);