#489310

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61479	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=hi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

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, 396506);