#9873187

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#40229	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9780383);