#9782787

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	#58197	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,e^2=g,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, 9689983);