#8720502

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

4006 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)(4,10)(5,14)(6,12)(7,9)(11,15)(13,16)(17,18,22,26)(19,21,25,23)(20,28,29,24)(27,30,32,31)(33,34,38,42)(35,37,41,39)(36,44,46,40)(43,45,48,47);(1,3)(2,8)(4,7)(5,13)(6,11)(9,10)(12,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(2,9)(3,4)(11,14)(12,13)(20,29)(24,28)(27,32)(30,31)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,20)(18,24)(19,27)(21,31)(22,29)(23,30)(25,32)(26,28)(33,35)(34,39)(36,43)(37,42)(38,41)(40,45)(44,47)(46,48);(1,5,7,15)(2,11,9,14)(3,13,4,12)(6,10,16,8)(17,19)(18,21)(20,27)(22,25)(23,26)(24,31)(28,30)(29,32)(33,36,38,46)(34,40,42,44)(35,43,41,48)(37,47,39,45);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,8,15,10)(17,21)(18,25)(19,26)(20,30)(22,23)(24,27)(28,32)(29,31)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,43,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627698);