#478111

order	512
nilpotency class	3
isoclinism	isoclinic class of #476106
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

702 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)(11,19)(12,16)(13,27)(14,18)(15,22)(17,28)(20,36)(21,31)(23,30)(24,34)(25,33)(26,35)(29,39)(32,37)(38,40);(1,3,6,2)(4,8,5,7)(9,11)(10,15)(12,20)(13,21)(14,23)(16,32)(17,30)(18,31)(19,27)(22,28)(24,37)(25,39)(26,38)(29,35)(33,36)(34,40);(1,4,6,5)(2,7,3,8)(9,12)(10,16)(11,20)(13,24)(14,26)(15,29)(17,33)(18,35)(19,36)(21,37)(22,39)(23,38)(25,28)(27,34)(30,40)(31,32);(1,5,6,4)(2,7,3,8)(11,21)(12,24)(13,28)(15,30)(16,33)(17,27)(19,31)(22,23)(25,26)(34,35)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=f,e^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,d^c=de,e^c=ei,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, 385307);