#8696756

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

4149 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603952);