#434084

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23438	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1147 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341280);