#490681

order	512
nilpotency class	4
isoclinism	isoclinic class of #490479
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

399 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)(17,18,23,28)(19,29,31,45)(20,25,35,43)(21,38,40,48)(22,41,39,27)(24,42,37,32)(26,44,34,30)(33,47,46,36);(17,19,30,25,22,33,40,42,23,31,44,43,39,46,21,32)(18,24,38,36,27,20,34,45,28,37,48,47,41,35,26,29);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,26)(21,36)(22,37)(23,35)(24,39)(27,42)(28,43)(29,44)(30,45)(31,34)(32,41)(33,38)(40,47)(46,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,31)(20,35)(24,37)(25,43)(29,45)(32,42)(33,46)(36,47)

Polycyclic presentation:

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

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, 397877);