#374515

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 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)(4,8)(5,6)(9,10,15,20,13,18,14,19)(11,21,24,36,23,16,30,32)(12,25,29,34,27,17,28,35)(22,31,26,33,38,37,39,40);(2,6)(7,8)(9,11)(10,16)(12,26)(13,23)(14,30)(15,24)(17,33)(18,21)(19,36)(20,32)(22,28)(25,40)(27,39)(29,38)(31,34)(35,37);(1,3,5,4)(2,7,6,8)(9,12,15,28,13,27,14,29)(10,17,19,35,18,25,20,34)(11,22,24,39,23,38,30,26)(16,31,36,33,21,37,32,40);(1,4,5,3)(2,7,6,8)(10,18)(16,21)(17,25)(19,20)(31,37)(32,36)(33,40)(34,35)

Polycyclic presentation:

2^9|a^2=h,c^2=efh,d^2=ei,g^2=i,h^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, 281711);