#8634567

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(33,34,38,39)(35,40,46,43)(36,37,42,41)(44,47,48,45);(1,3,7,6)(2,5,8,10)(4,13,15,11)(9,16,12,14)(17,19)(18,24)(20,27)(21,22)(23,26)(25,31)(28,30)(29,32)(33,34,38,39)(35,43,46,40)(36,37,42,41)(44,45,48,47);(2,8)(5,10)(9,12)(14,16)(19,26)(21,24)(27,32)(28,31)(34,39)(36,42)(40,43)(44,48);(1,4,7,15)(2,9,8,12)(3,11,6,13)(5,14,10,16)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,35)(34,40)(36,44)(37,47)(38,46)(39,43)(41,45)(42,48);(1,5)(2,3)(4,14)(6,8)(7,10)(9,11)(12,13)(15,16)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32)(33,36,38,42)(34,41,39,37)(35,44,46,48)(40,45,43,47);(1,6,7,3)(2,5,8,10)(4,11,15,13)(9,16,12,14)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,37,38,41)(34,42,39,36)(35,45,46,47)(40,44,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=gh,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, 8541763);