#459348

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
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

845 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366544);