#8463210

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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gh,d^2=h,e^2=ghi,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, 8370406);