#8975302

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

4061 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8882498);