#8533623

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

4299 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:

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

Polycyclic presentation:

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

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, 8440819);