#8115409

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3781 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)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(2,3)(4,11)(8,10)(12,16)(18,22)(19,26)(24,28)(27,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=g,f^2=gi,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, 8022605);