#8639725

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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)(3,8)(4,5)(6,7)(9,14)(10,13)(11,15)(12,16)(18,24)(19,20)(21,29)(31,32)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(1,2)(3,10)(4,5)(6,7)(8,13)(9,14)(11,12)(15,16)(17,18)(19,25)(20,22)(21,30)(23,24)(26,29)(27,31)(28,32)(35,41)(39,43)(42,47)(46,48);(2,7)(4,14)(8,10)(11,16)(17,19,23,20)(18,25,24,22)(21,28,29,27)(26,31,30,32)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,3)(2,8)(4,11)(5,15)(6,13)(7,10)(9,12)(14,16)(17,20,23,19)(18,25,24,22)(21,28,29,27)(26,32,30,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,4)(2,9)(3,11)(5,7)(6,14)(8,12)(10,15)(13,16)(17,21)(18,26)(19,27)(20,28)(22,31)(23,29)(24,30)(25,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,5,6,9)(2,4,7,14)(3,12,13,15)(8,16,10,11)(17,22,23,25)(18,20,24,19)(21,31,29,32)(26,28,30,27)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8546921);