#8202266

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

3846 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(34,39)(35,40)(41,46)(42,47);(1,2,7,4)(3,8,6,10)(5,9,14,12)(11,15,13,16)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109462);