#9478150

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	#57573	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=ghi,e^2=ghi,f^2=gh,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, 9385346);