#9863333

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

4095 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)(3,9)(4,13)(5,14)(6,8)(7,12)(10,11)(15,16)(17,18,22,27)(19,23,30,24)(20,25,31,29)(21,28,32,26)(34,40)(36,43)(41,47)(44,46);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,34)(35,41)(36,42)(37,47)(38,43)(39,40)(44,48)(45,46);(2,8)(4,14)(7,9)(10,16)(17,19,22,30)(18,24,27,23)(20,28,31,26)(21,29,32,25)(33,35,39,37)(34,41,40,47)(36,44,43,46)(38,45,42,48);(1,4)(2,5)(3,10)(6,14)(7,11)(8,13)(9,15)(12,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,36,39,43)(34,42,40,38)(35,44,37,46)(41,48,47,45);(1,3)(2,9)(4,10)(5,15)(6,12)(7,8)(11,13)(14,16)(17,20)(18,25)(19,28)(21,24)(22,31)(23,32)(26,30)(27,29)(33,37,39,35)(34,41,40,47)(36,46,43,44)(38,45,42,48);(1,5)(2,4)(3,11)(6,13)(7,10)(8,14)(9,16)(12,15)(17,21,22,32)(18,26,27,28)(19,29,30,25)(20,23,31,24)(33,38,39,42)(34,43,40,36)(35,45,37,48)(41,46,47,44)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=ghi,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, 9770529);