#8184748

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,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=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8091944);