#389230

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2358
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=efh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296426);