#8549791

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

4092 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,f^2=i,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, 8456987);