#9545557

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	#54190	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9452753);