#9947000

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4127 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)(17,18,23,24)(19,22,20,27)(21,25,30,32)(26,31,28,29)(33,34,38,40)(35,39,43,46)(36,37,44,45)(41,42,47,48);(2,3)(7,9)(8,10)(15,16)(17,19)(18,22)(20,23)(21,26)(24,27)(25,31)(28,30)(29,32)(33,35,38,43)(34,39,40,46)(36,41,44,47)(37,42,45,48);(2,3)(7,9)(8,10)(15,16)(18,24)(19,20)(25,32)(26,28)(34,40)(36,44)(39,46)(41,47);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,20,23,19)(18,22,24,27)(21,28,30,26)(25,31,32,29)(33,36)(34,37)(35,41)(38,44)(39,42)(40,45)(43,47)(46,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,25)(19,26)(20,28)(22,31)(23,30)(24,32)(27,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,22)(18,19)(20,24)(21,29)(23,27)(25,28)(26,32)(30,31)(33,37)(34,36)(35,42)(38,45)(39,41)(40,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gi,e^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, 9854196);