#9711921

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	#55440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,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, 9619117);