#8533130

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

4272 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440326);