#8425523

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

4194 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332719);