#489108

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61290	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

680 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)(3,11,14,15)(4,9,5,8)(7,13,16,12)(18,23)(20,30)(21,33)(24,39)(25,38)(26,42)(28,44)(29,43)(31,35)(32,47)(34,36)(37,40)(41,48)(45,46);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,18,22,25)(19,24,27,40)(20,31,34,42)(21,35,33,26)(23,36,38,30)(28,45,32,48)(29,46,43,41)(37,44,39,47);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,19)(18,24)(20,32)(21,29)(22,27)(23,37)(25,40)(26,41)(28,34)(30,47)(31,48)(33,43)(35,46)(36,44)(38,39)(42,45);(19,27)(24,40)(28,32)(29,43)(37,39)(41,46)(44,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396304);