#496441

order	512
nilpotency class	4
isoclinism	isoclinic class of #481373
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

763 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)(17,18)(19,27,28,43)(20,23)(21,33)(22,36)(24,29,38,45)(25,35)(26,32)(30,41,37,44)(31,47)(34,42,46,40)(39,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(18,23)(19,28)(25,39)(26,36)(29,45)(30,37)(33,47)(34,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,20,28)(18,24,23,38)(21,34,31,46)(22,37,32,30)(25,41,39,44)(26,42,36,40)(27,33,43,47)(29,35,45,48);(2,8)(6,9)(7,16)(14,15)(18,23)(24,38)(25,39)(26,36)(27,43)(33,47)(40,42)(41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,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,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403637);