#9850781

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	#38512	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(2,4)(7,9)(8,12)(14,15)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(18,25)(19,20)(26,32)(27,28)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,4,6,2)(3,9,11,7)(5,8,13,12)(10,14,16,15)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=g,e^2=gi,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, 9757977);