#8621889

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

4182 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(18,22)(23,25)(24,28)(30,31)(33,34,38,39)(35,40,46,43)(36,37,42,41)(44,47,48,45);(2,7)(3,11)(4,14)(13,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,34,38,39)(35,43,46,40)(36,37,42,41)(44,45,48,47);(2,7)(4,14)(8,10)(12,16)(19,26)(23,25)(27,32)(30,31)(34,39)(36,42)(40,43)(44,48);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,35)(34,40)(36,44)(37,47)(38,46)(39,43)(41,45)(42,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,36,38,42)(34,41,39,37)(35,44,46,48)(40,45,43,47);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,41)(34,42,39,36)(35,45,46,47)(40,44,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529085);