#9730306

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	#49219	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,25)(19,24,22,32)(20,26,29,21)(27,28,30,31)(33,34,38,43)(35,39,46,40)(36,41,42,37)(44,45,47,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,22)(18,24,25,32)(20,27,29,30)(21,31,26,28)(33,35)(34,39)(36,47)(37,45)(38,46)(40,43)(41,48)(42,44);(2,3)(7,9)(8,10)(14,15)(18,25)(20,29)(24,32)(27,30)(33,35,38,46)(34,40,43,39)(36,44,42,47)(37,45,41,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,26)(19,27)(21,25)(22,30)(23,29)(24,28)(31,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,48)(42,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,26)(18,20,25,29)(19,28,22,31)(24,30,32,27)(33,36)(34,41)(35,44)(37,43)(38,42)(39,45)(40,48)(46,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,19)(18,24,25,32)(20,30,29,27)(21,31,26,28)(33,37,38,41)(34,42,43,36)(35,45,46,48)(39,44,40,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9637502);