#9214735

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	#37038	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9121931);