#389270

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	#2358
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

802 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)(9,10)(11,16)(12,25,27,34)(13,28,30,38)(14,19)(15,20)(17,21,39,31)(18,26,33,29)(22,35)(23,36)(24,37)(32,40);(1,3,6,2)(4,8,5,7)(9,11)(10,16)(12,26)(13,21)(14,22)(15,24)(17,38)(18,34)(19,35)(20,37)(23,32)(25,33)(27,29)(28,39)(30,31)(36,40);(1,4,6,5)(2,7,3,8)(9,12,14,31,15,27,32,21)(10,17,19,34,20,39,40,25)(11,13,22,26,24,30,23,29)(16,33,35,38,37,18,36,28);(1,5,6,4)(2,7,3,8)(9,13)(10,18)(11,21)(12,22)(14,26)(15,30)(16,34)(17,36)(19,28)(20,33)(23,27)(24,31)(25,37)(29,32)(35,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=efh,d^2=e,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, 296466);