#491529

order	512
nilpotency class	4
isoclinism	isoclinic class of #491511
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

527 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:

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

Polycyclic presentation:

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

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, 398725);