#374596

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281792);