#496442

order	512
nilpotency class	4
isoclinism	isoclinic class of #481393
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

718 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(19,27,28,44)(20,30)(21,33)(23,35,36,47)(25,40)(26,39)(29,31,34,45)(32,43)(37,41,42,48)(38,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,35)(28,36)(29,37)(30,40)(31,41)(32,38)(33,39)(34,42)(43,46)(44,47)(45,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,23)(20,31)(21,34)(22,28)(24,36)(25,41)(26,42)(27,43)(29,33)(30,44)(32,45)(35,46)(37,39)(38,48)(40,47);(2,8)(6,9)(7,16)(14,15)(18,24)(23,36)(25,38)(26,39)(35,47)(37,42)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fhi,g^c=gi,h^c=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, 403638);