#8605714

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(3,10)(4,12)(5,6)(7,8)(9,15)(11,14)(13,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,32)(27,30)(28,31)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3,7,6)(2,5,8,10)(4,13,15,11)(9,16,12,14)(17,19,23,27)(18,20,24,30)(21,31,32,22)(25,26,29,28)(34,35)(39,41)(40,42)(46,47);(2,8)(5,10)(9,12)(14,16)(18,24)(19,27)(21,32)(26,28)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,4,7,15)(2,9,8,12)(3,11,6,13)(5,14,10,16)(17,20)(18,19)(21,28)(22,25)(23,30)(24,27)(26,32)(29,31)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5)(2,3)(4,14)(6,8)(7,10)(9,11)(12,13)(15,16)(17,21)(18,25)(19,22)(20,28)(23,32)(24,29)(26,30)(27,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,6,7,3)(2,5,8,10)(4,11,15,13)(9,16,12,14)(17,22)(18,26)(19,21)(20,29)(23,31)(24,28)(25,30)(27,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=g,f^2=g,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, 8512910);