#8202223

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

4130 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,15)(14,16)(18,19)(23,25)(24,26)(30,31)(33,34,39,41)(35,40,44,48)(36,38,43,46)(37,45,47,42);(2,7)(3,10)(4,13)(14,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,39,44)(34,40,41,48)(36,42,43,45)(37,46,47,38);(2,7)(5,8)(9,12)(14,15)(21,29)(24,26)(27,32)(30,31)(34,41)(35,44)(38,46)(42,45);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,39,43)(34,38,41,46)(35,45,44,42)(37,40,47,48);(1,4,6,13)(2,9,7,12)(3,11,10,16)(5,14,8,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37)(34,42)(35,38)(36,40)(39,47)(41,45)(43,48)(44,46);(1,5,6,8)(2,3,7,10)(4,14,13,15)(9,11,12,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38)(34,43)(35,37)(36,41)(39,46)(40,45)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=g,f^2=g,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, 8109419);