#399500

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

816 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306696);