#8202458

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

4075 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,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, 8109654);