#478132

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

703 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=f,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=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385328);