#8334484

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

4043 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,13)(14,16)(17,18,23,20)(19,22,25,26)(21,24,30,28)(27,29,31,32)(33,34)(35,36)(37,45)(38,39)(40,47)(41,44)(42,46)(43,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,38,41)(34,36,39,44)(37,42,47,48)(40,43,45,46);(2,9)(6,10)(8,11)(13,16)(19,25)(22,26)(27,31)(29,32)(34,39)(35,41)(40,45)(42,48);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,13)(11,16)(17,20,23,18)(19,26,25,22)(21,24,30,28)(27,29,31,32)(33,36)(34,35)(37,46)(38,44)(39,41)(40,48)(42,45)(43,47);(1,5,7,3)(2,11,9,8)(4,12,15,14)(6,16,10,13)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,40)(35,42)(36,43)(38,47)(39,45)(41,48)(44,46);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=ghi,e^2=g,f^2=hi,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, 8241680);