#490673

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
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	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

407 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,22,27)(19,28,33,44)(20,25,23,35)(21,37,38,46)(24,39,30,32)(26,36,41,29)(31,42,34,48)(40,45,43,47);(17,19,29,45,23,34,38,39,22,33,36,47,20,31,21,32)(18,24,37,42,25,40,41,44,27,30,46,48,35,43,26,28);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,18)(19,30)(20,35)(21,26)(22,27)(23,25)(24,33)(28,39)(29,37)(31,40)(32,44)(34,43)(36,46)(38,41)(42,47)(45,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,23)(18,25,27,35)(19,31,33,34)(21,36,38,29)(24,40,30,43)(26,37,41,46)(28,42,44,48)(32,47,39,45)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,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,c^b=ci,d^c=di,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397869);