#483997

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#466	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=dhi,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

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, 391193);