#8323410

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8230606);