#8095012

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

4232 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002208);