#8621615

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

4162 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18,21,24)(19,22,26,25)(20,23,29,28)(27,30,32,31)(33,34)(35,40)(36,37)(38,43)(39,41)(42,47)(44,48)(45,46);(2,4)(3,10)(8,12)(11,13)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,35,39,44)(34,40,41,48)(36,45,42,43)(37,46,47,38);(2,4)(7,9)(8,12)(15,16)(19,26)(22,25)(27,32)(30,31)(34,41)(35,44)(37,47)(43,45);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,31)(30,32)(33,36,39,42)(34,37,41,47)(35,43,44,45)(38,48,46,40);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,37)(34,42)(35,38)(36,41)(39,47)(40,45)(43,48)(44,46);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,38)(34,43)(35,37)(36,40)(39,46)(41,45)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528811);