#8174666

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

4049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081862);