#9066036

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	#41342	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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)(3,8,12,14)(4,6,10,16)(5,9,15,13)(17,18,23,26)(19,24,29,21)(20,22,32,25)(27,31,30,28)(33,34,39,37)(35,36,42,40)(38,45,46,41)(43,47,48,44);(1,3)(2,8)(4,13)(5,6)(7,12)(9,10)(11,14)(15,16)(17,19,23,29)(18,24,26,21)(20,30,32,27)(22,28,25,31)(34,37)(35,42)(41,45)(43,48);(3,12)(5,15)(8,14)(9,13)(17,20,23,32)(18,25,26,22)(19,27,29,30)(21,31,24,28)(33,35,39,42)(34,40,37,36)(38,43,46,48)(41,47,45,44);(1,4)(2,6)(3,13)(5,8)(7,10)(9,12)(11,16)(14,15)(17,21)(18,19)(20,31)(22,30)(23,24)(25,27)(26,29)(28,32)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,5)(2,9)(3,6)(4,14)(7,15)(8,10)(11,13)(12,16)(17,20)(18,22)(19,27)(21,28)(23,32)(24,31)(25,26)(29,30)(33,37,39,34)(35,36,42,40)(38,45,46,41)(43,44,48,47);(1,6)(2,10)(3,5)(4,11)(7,16)(8,9)(12,15)(13,14)(17,22)(18,20)(19,28)(21,27)(23,25)(24,30)(26,32)(29,31)(33,38,39,46)(34,41,37,45)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,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^c=egi,e^d=egi,f^d=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, 8973232);