#8463863

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

4039 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371059);