#8211652

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3824 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8118848);