#387902

order	512
nilpotency class	3
isoclinism	isoclinic class of #232128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

856 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)(11,20,24,16)(12,25,29,40)(13,18)(14,30)(15,19)(17,27,34,28)(21,37)(22,35,23,32)(26,36)(31,38)(33,39);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,26)(13,23)(14,22)(15,24)(17,33)(18,35)(19,20)(21,28)(25,31)(27,39)(29,38)(30,32)(34,37)(36,40);(1,4,5,8)(2,6,3,7)(9,12,14,28,15,29,13,27)(10,17,18,25,19,34,30,40)(11,21,23,26,24,39,22,38)(16,31,32,33,20,36,35,37);(2,3)(6,7)(10,18,19,30)(16,32,20,35)(17,25,34,40)(31,33,36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=efi,d^2=fg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^c=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, 295098);