#8745933

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

4100 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)(3,4)(5,15)(6,11)(7,9)(8,13)(10,12)(14,16)(17,18,22,24)(19,23,27,28)(20,29,30,25)(21,26,32,31)(33,34)(35,41)(36,40)(37,42)(38,43)(39,46)(44,45)(47,48);(1,3,7,13)(2,8,9,4)(5,6,12,16)(10,11,15,14)(17,19)(18,23)(20,26)(21,25)(22,27)(24,28)(29,32)(30,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(2,9)(4,8)(10,15)(11,14)(18,24)(19,27)(25,29)(26,31)(33,36,38,42)(34,40,43,37)(35,44,46,48)(39,47,41,45);(1,4,7,8)(2,3,9,13)(5,14,12,11)(6,15,16,10)(17,18)(19,28)(20,25)(21,31)(22,24)(23,27)(26,32)(29,30)(33,36)(34,40)(35,44)(37,43)(38,42)(39,47)(41,45)(46,48);(1,5)(2,10)(3,6)(4,14)(7,12)(8,11)(9,15)(13,16)(17,20)(18,25)(19,26)(21,23)(22,30)(24,29)(27,31)(28,32)(33,35)(34,41)(36,44)(37,47)(38,46)(39,43)(40,45)(42,48);(1,6)(2,11)(3,12)(4,10)(5,13)(7,16)(8,15)(9,14)(17,21,22,32)(18,26,24,31)(19,25,27,29)(20,28,30,23)(33,37,38,40)(34,42,43,36)(35,45,46,47)(39,48,41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=hi,f^2=gh,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, 8653129);