#8276814

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

4099 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)(3,11)(4,14)(5,10)(6,8)(7,13)(9,15)(12,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(35,43)(37,48)(40,44)(41,46);(1,3,6,13)(2,7,8,11)(4,12,15,10)(5,9,16,14)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,34,39,42)(35,44,43,40)(36,47,45,38)(37,41,48,46);(2,8)(5,16)(7,11)(9,14)(19,26)(23,30)(27,31)(28,32)(33,35,39,43)(34,40,42,44)(36,37,45,48)(38,41,47,46);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,15)(14,16)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36)(34,38)(35,37)(39,45)(40,41)(42,47)(43,48)(44,46);(1,4)(2,9)(3,12)(5,11)(6,15)(7,16)(8,14)(10,13)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37)(34,41)(35,45)(36,43)(38,44)(39,48)(40,47)(42,46);(1,5,6,16)(2,10,8,12)(3,9,13,14)(4,11,15,7)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,38)(34,36)(35,46)(37,44)(39,47)(40,48)(41,43)(42,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8184010);