#8180382

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

4153 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087578);