#383398

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,25,13,17)(14,18)(15,19)(16,23)(21,31,40,39)(22,34,27,32)(24,37)(26,38,28,35)(29,36)(30,33);(1,3,5,2)(4,7,8,6)(9,11,14,23)(10,16,18,20)(12,22,13,27)(15,30,29,24)(17,34,25,32)(19,37,36,33)(21,28,40,26)(31,38,39,35);(1,4,5,8)(2,6,3,7)(9,12,14,13)(10,17,18,25)(11,21,23,40)(15,28,29,26)(16,31,20,39)(19,38,36,35)(22,24,27,30)(32,37,34,33);(2,3)(6,7)(9,13,14,12)(10,17,18,25)(11,22,23,27)(15,28,29,26)(16,32,20,34)(19,35,36,38)(21,24,40,30)(31,37,39,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^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,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290594);