#8139622

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

4233 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)(3,10)(4,8)(5,15)(6,16)(7,9)(11,12)(13,14)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(37,45)(40,47)(42,48)(44,46);(3,4)(8,10)(11,13)(12,14)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,47)(46,48);(1,3,7,4)(2,8,9,10)(5,11,16,13)(6,12,15,14)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4,7,3)(2,8,9,10)(5,11,16,13)(6,14,15,12)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,36,38,39)(34,35,41,43)(37,44,45,46)(40,42,47,48);(1,5)(2,6)(3,11)(4,13)(7,16)(8,12)(9,15)(10,14)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,6)(2,5)(3,12)(4,14)(7,15)(8,11)(9,16)(10,13)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046818);