#9851091

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	#53961	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=h,e^2=i,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, 9758287);