#8873725

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

3954 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)(3,9)(4,12)(5,10)(6,7)(8,13)(11,15)(14,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(2,7)(4,13)(5,9)(11,14)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(2,7)(3,10)(8,12)(11,14)(18,19)(23,25)(24,26)(30,31)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,3)(2,5)(4,14)(6,10)(7,9)(8,16)(11,13)(12,15)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,4)(2,8)(3,11)(5,15)(6,13)(7,12)(9,16)(10,14)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,5)(2,3)(4,15)(6,9)(7,10)(8,11)(12,14)(13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

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

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, 8780921);