#8270618

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

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

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, 8177814);