#8505384

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

4110 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412580);