#8622225

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

4086 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)(4,12)(5,6)(7,8)(9,15)(11,14)(13,16)(18,23)(19,25)(24,27)(26,31)(33,34,39,41)(35,40,45,48)(36,42,43,37)(38,44,47,46);(1,3,7,6)(2,5,8,10)(4,13,15,11)(9,16,12,14)(17,18,22,23)(19,21,25,30)(20,27,28,24)(26,32,31,29)(33,35,39,45)(34,40,41,48)(36,44,43,46)(37,38,42,47);(2,8)(5,10)(9,12)(14,16)(20,28)(24,27)(26,31)(29,32)(34,41)(35,45)(37,42)(44,46);(1,4,7,15)(2,9,8,12)(3,11,6,13)(5,14,10,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,32)(28,31)(33,36,39,43)(34,42,41,37)(35,46,45,44)(38,48,47,40);(1,5)(2,3)(4,14)(6,8)(7,10)(9,11)(12,13)(15,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,27)(25,31)(30,32)(33,37,39,42)(34,43,41,36)(35,47,45,38)(40,44,48,46);(1,6,7,3)(2,5,8,10)(4,11,15,13)(9,16,12,14)(17,21)(18,25)(19,23)(20,29)(22,30)(24,31)(26,27)(28,32)(33,38,39,47)(34,44,41,46)(35,42,45,37)(36,40,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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, 8529421);