#8110789

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

4346 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(20,28)(24,29)(26,32)(30,31)(34,38)(35,41)(36,44)(46,48);(2,6)(7,9)(8,12)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(3,10)(7,9)(11,16)(14,15)(18,22)(23,25)(24,29)(30,31)(33,34,37,38)(35,39,41,42)(36,40,44,45)(43,46,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017985);