#8112453

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,e^2=h,f^2=gh,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, 8019649);