#8360885

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	#55054	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=i,e^2=gh,f^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,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, 8268081);