#8630797

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

4160 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=gh,e^2=i,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, 8537993);