#491428

order	512
nilpotency class	4
isoclinism	isoclinic class of #491239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19222	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

412 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=egi,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=dhi,d^c=di,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398624);