#390223

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

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	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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)(3,6)(4,7)(5,8)(10,16,19,20)(11,21)(12,25,27,28)(17,31,33,22)(23,37,35,38)(24,29)(32,40)(36,39);(1,3,5,4)(2,6,8,7)(9,10,14,19)(11,22,21,31)(12,26,27,30)(13,20,15,16)(17,29,33,24)(18,28,34,25)(23,32,35,40)(36,37,39,38);(1,3,5,4)(2,7,8,6)(9,11,15,29,14,21,13,24)(10,17,16,22,19,33,20,31)(12,23,25,38,27,35,28,37)(18,32,30,36,34,40,26,39);(1,4,5,3)(2,6,8,7)(9,12,14,27)(10,18,19,34)(11,23,21,35)(13,28,15,25)(16,30,20,26)(17,32,33,40)(22,36,31,39)(24,37,29,38)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=efh,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,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, 297419);