#8818617

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

4242 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725813);