#478131

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,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, 385327);