#9782682

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

4232 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,23,25)(19,24,26,21)(20,22,29,32)(27,28,31,30)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,26)(18,24,25,21)(20,27,29,31)(22,28,32,30)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,7)(6,13)(8,11)(14,15)(18,25)(19,26)(20,29)(28,30)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20)(18,22)(19,27)(21,30)(23,29)(24,28)(25,32)(26,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,19)(20,30)(22,27)(23,24)(25,26)(28,29)(31,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,48)(41,47)(43,44);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22)(18,20)(19,28)(21,31)(23,32)(24,27)(25,29)(26,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,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, 9689878);