#8114394

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

3971 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021590);