#408195

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

969 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)(3,8)(4,6)(5,7)(9,10,14,19)(11,21,24,17)(12,27,30,36)(13,16,20,15)(18,31,35,29)(22,34,40,38)(23,25,33,26)(28,32,39,37);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,17,19,21)(12,28,30,39)(13,23,20,33)(15,25,16,26)(18,34,35,38)(22,31,40,29)(27,32,36,37);(9,12)(10,18)(11,22)(13,27)(14,30)(15,31)(16,29)(17,32)(19,35)(20,36)(21,37)(23,38)(24,40)(25,28)(26,39)(33,34);(1,4,5,3)(2,6,7,8)(9,13,16,19,14,20,15,10)(11,23,26,21,24,33,25,17)(12,27,29,35,30,36,31,18)(22,38,39,37,40,34,28,32)

Polycyclic presentation:

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

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, 315391);