#9015359

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3800 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)(3,8)(4,10)(5,11)(6,7)(9,13)(12,16)(14,15)(17,18,23,21)(19,24,29,27)(20,25,31,22)(26,28,30,32)(33,34,39,44)(35,45,47,40)(36,43,38,41)(37,42,46,48);(1,3,7,15)(2,8,6,14)(4,12,13,5)(9,11,10,16)(17,19)(18,24)(20,30)(21,27)(22,28)(23,29)(25,32)(26,31)(33,34)(35,45)(36,41)(37,48)(38,43)(39,44)(40,47)(42,46);(1,4,7,13)(2,9,6,10)(3,5,15,12)(8,16,14,11)(17,20,23,31)(18,25,21,22)(19,26,29,30)(24,28,27,32)(33,35,39,47)(34,40,44,45)(36,46,38,37)(41,48,43,42);(1,4)(2,10)(3,12)(5,15)(6,9)(7,13)(8,16)(11,14)(17,21,23,18)(19,27,29,24)(20,25,31,22)(26,28,30,32)(33,36,39,38)(34,41,44,43)(35,46,47,37)(40,48,45,42);(1,5,7,12)(2,11,6,16)(3,13,15,4)(8,9,14,10)(17,22,23,25)(18,20,21,31)(19,28,29,32)(24,30,27,26)(33,37)(34,42)(35,36)(38,47)(39,46)(40,41)(43,45)(44,48);(1,6,7,2)(3,14,15,8)(4,9,13,10)(5,16,12,11)(17,19)(18,24)(20,30)(21,27)(22,28)(23,29)(25,32)(26,31)(33,38,39,36)(34,43,44,41)(35,37,47,46)(40,42,45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922555);