#8505519

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

4133 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)(3,9)(4,13)(5,16)(6,10)(7,8)(11,15)(12,14)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34)(35,40)(36,37)(38,42)(39,43)(41,47)(44,48)(45,46);(2,8)(5,10)(9,13)(11,14)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(33,35,39,45)(34,40,43,46)(36,44,47,42)(37,48,41,38);(1,3,7,4)(2,9,8,13)(5,11,10,14)(6,12,16,15)(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,36)(34,41)(35,42)(37,43)(38,46)(39,47)(40,48)(44,45);(1,4,7,3)(2,9,8,13)(5,14,10,11)(6,12,16,15)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,37,39,41)(34,36,43,47)(35,38,45,48)(40,42,46,44);(1,5,7,10)(2,6,8,16)(3,11,4,14)(9,12,13,15)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,38)(34,42)(35,41)(36,46)(37,45)(39,48)(40,47)(43,44);(1,6)(2,10)(3,12)(4,15)(5,8)(7,16)(9,14)(11,13)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,36)(34,37)(35,44)(38,46)(39,47)(40,48)(41,43)(42,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=gi,e^2=g,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, 8412715);