#9127350

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	#51637	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034546);