#386187

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	D₄×C₂²⋊C₄
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1159 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)(4,7)(6,8)(11,20,24,23)(12,25,27,26)(15,21,30,33)(16,31,32,34);(1,3)(2,4)(5,8)(6,7)(9,10)(11,21)(12,16)(13,17)(14,19)(15,23)(18,28)(20,30)(22,36)(24,33)(25,31)(26,34)(27,32)(29,37)(35,40)(38,39);(1,4,6,5)(2,7,8,3)(9,11,13,23,14,24,28,20)(10,15,18,21,19,30,17,33)(12,22,26,37,27,38,25,35)(16,29,31,36,32,40,34,39);(1,5,6,4)(2,7,8,3)(9,12)(10,16)(11,22)(13,26)(14,27)(15,29)(17,34)(18,31)(19,32)(20,35)(21,36)(23,37)(24,38)(25,28)(30,40)(33,39)

Polycyclic presentation:

2^9|a^2=h,c^2=eg,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293383);