#9938082

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

4027 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,36)(37,40)(38,46)(39,41)(42,47)(43,48)(44,45);(1,2,5,8)(3,9,11,6)(4,7,13,16)(10,15,12,14)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35,39,37)(34,40,41,36)(38,43,47,45)(42,48,46,44);(18,25)(19,20)(26,32)(27,28)(34,41)(36,40)(42,46)(44,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36,39,40)(34,35,41,37)(38,44,47,48)(42,43,46,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,37,39,35)(34,40,41,36)(38,43,47,45)(42,44,46,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,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, 9845278);