#8202413

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

4071 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)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18,24,28)(19,29,31,25)(20,21,22,26)(23,27,30,32)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(3,4)(5,14)(7,10)(8,13)(17,19)(18,25)(20,27)(21,23)(22,32)(24,31)(26,30)(28,29)(34,39)(35,42)(41,44)(43,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,20,24,22)(18,26,28,21)(19,27,31,32)(23,25,30,29)(33,34,38,39)(35,40,42,37)(36,41,46,44)(43,47,48,45);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21)(18,20)(19,30)(22,28)(23,31)(24,26)(25,32)(27,29)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22,24,20)(18,21,28,26)(19,27,31,32)(23,25,30,29)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,23,24,30)(18,27,28,32)(19,21,31,26)(20,29,22,25)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=gh,e^2=g,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, 8109609);