#432576

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

965 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,33)(28,35)(29,44)(30,39)(31,40)(34,41)(37,42)(38,46)(43,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,38)(26,28)(27,40)(32,44)(33,46)(34,47)(36,39)(37,45)(41,43)(42,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,18)(19,28)(20,25)(21,36)(22,26)(23,27)(24,35)(29,43)(30,40)(31,39)(32,33)(34,42)(37,41)(38,46)(44,47)(45,48);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,20,23,34)(18,25,27,42)(19,29,31,47)(21,33,22,37)(24,38,40,48)(26,41,36,32)(28,43,39,44)(30,45,35,46)

Polycyclic presentation:

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

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, 339772);