#9863327

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

4103 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)(3,11)(4,7)(5,9)(6,15)(8,13)(10,16)(12,14)(17,18,22,27)(19,23,30,24)(20,25,31,29)(21,28,32,26)(34,40)(36,45)(41,47)(43,46);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,10,14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,34,39,40)(35,41,37,47)(36,42,45,38)(43,48,46,44);(2,4)(8,11)(9,14)(10,15)(17,19,22,30)(18,24,27,23)(20,28,31,26)(21,29,32,25)(33,35,39,37)(34,41,40,47)(36,43,45,46)(38,44,42,48);(1,4,7,2)(3,11,13,8)(5,14,12,9)(6,15,16,10)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,36)(34,42)(35,43)(37,46)(38,40)(39,45)(41,48)(44,47);(1,5)(2,9)(3,6)(4,14)(7,12)(8,10)(11,15)(13,16)(17,20)(18,25)(19,28)(21,24)(22,31)(23,32)(26,30)(27,29)(33,37,39,35)(34,41,40,47)(36,46,45,43)(38,44,42,48);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,14)(9,11)(17,21,22,32)(18,26,27,28)(19,29,30,25)(20,23,31,24)(33,38,39,42)(34,36,40,45)(35,44,37,48)(41,43,47,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=hi,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, 9770523);