#8463201

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=hi,f^2=gi,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, 8370397);