#9794361

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	#54942	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4261 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)(17,18,22,24)(19,25,29,23)(20,30,26,21)(27,28,32,31)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,23)(20,27)(21,28)(22,29)(24,25)(26,32)(30,31)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(2,3)(7,9)(8,10)(14,15)(18,24)(20,26)(23,25)(27,32)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,26)(18,21,24,30)(19,27,29,32)(23,28,25,31)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19)(18,25)(20,27)(21,31)(22,29)(23,24)(26,32)(28,30)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,26)(19,28)(20,24)(22,30)(23,32)(25,27)(29,31)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 9701557);