#9947203

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	#56325	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 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)(4,7,11,9)(5,8,14,10)(12,15,13,16)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32)(33,34)(35,39)(36,41)(37,42)(38,40)(43,44)(45,47)(46,48);(2,3)(4,11)(8,10)(12,13)(20,27)(24,30)(26,31)(28,32)(33,35,38,42)(34,39,40,37)(36,43,45,48)(41,44,47,46);(18,23)(19,25)(24,30)(26,31)(34,40)(35,42)(41,47)(43,48);(1,3,6,2)(4,9,11,7)(5,10,14,8)(12,16,13,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,35)(34,37)(36,43)(38,42)(39,40)(41,46)(44,47)(45,48);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,36)(34,41)(35,43)(37,46)(38,45)(39,44)(40,47)(42,48);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37)(34,35)(36,44)(38,39)(40,42)(41,48)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854399);