#9947125

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,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, 9854321);