#8622266

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

4090 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,21,24)(19,22,26,25)(20,23,29,28)(27,30,32,31)(33,34)(35,40)(36,41)(37,39)(38,42)(43,46)(44,48)(45,47);(2,7)(3,11)(4,14)(13,15)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,35,39,42)(34,40,37,38)(36,44,47,43)(41,48,45,46);(2,7)(4,14)(8,10)(12,16)(19,26)(22,25)(27,32)(30,31)(34,37)(35,42)(41,45)(43,44);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,31)(30,32)(33,36,39,47)(34,41,37,45)(35,43,42,44)(38,48,40,46);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,37,39,34)(35,40,42,38)(36,45,47,41)(43,46,44,48);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,38,39,40)(34,42,37,35)(36,46,47,48)(41,43,45,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=h,e^2=i,f^2=h,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, 8529462);