#9847637

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	#52353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,e^2=ghi,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, 9754833);