#9245136

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	#51130	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9152332);