#8850607

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	#55094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8757803);