#8843420

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

3943 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8750616);