#9206595

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

3937 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9113791);