#8188010

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

4145 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095206);