#9167682

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	#51687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=gh,f^2=g,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, 9074878);