#8730905

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

4096 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8638101);