#8728558

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

4003 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(35,37)(36,40)(43,45)(44,46);(2,6)(7,9)(8,12)(15,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,34,39,42)(35,36,37,40)(38,41,47,48)(43,44,45,46);(2,6)(7,9)(8,12)(15,16)(19,26)(22,23)(27,32)(30,31)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36,39,40)(34,35,42,37)(38,46,47,44)(41,45,48,43);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,48,46);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8635754);