#9447352

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	#58087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,21,23)(19,22,25,30)(20,24,28,31)(26,29,27,32)(33,34,39,43)(35,38,45,40)(36,46,44,41)(37,47,48,42);(3,9)(7,13)(10,15)(14,16)(17,19)(18,22)(20,27)(21,25)(23,30)(24,32)(26,28)(29,31)(33,35,39,45)(34,40,43,38)(36,37,44,48)(41,47,46,42);(2,6)(7,13)(8,11)(14,16)(18,23)(19,25)(24,31)(26,27)(33,36,39,44)(34,41,43,46)(35,37,45,48)(38,42,40,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,22)(20,27)(21,25)(23,30)(24,32)(26,28)(29,31)(33,37,39,48)(34,42,43,47)(35,44,45,36)(38,46,40,41);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,21,28)(18,24,23,31)(19,26,25,27)(22,29,30,32)(33,35)(34,38)(36,37)(39,45)(40,43)(41,42)(44,48)(46,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,32)(28,31)(33,38)(34,35)(36,47)(37,46)(39,40)(41,48)(42,44)(43,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9354548);