#8724812

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(3,7)(4,12)(5,10)(6,8)(9,14)(11,15)(13,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,39,41)(35,43,37,40)(36,46,47,42)(38,44,48,45);(1,3,6,10)(2,7,8,5)(4,11,14,16)(9,13,12,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35,39,37)(34,40,41,43)(36,44,47,45)(38,42,48,46);(2,8)(3,10)(9,12)(11,16)(20,29)(24,28)(26,32)(30,31)(34,41)(36,47)(40,43)(44,45);(1,3)(2,7)(4,11)(5,8)(6,10)(9,13)(12,15)(14,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,36,39,47)(34,42,41,46)(35,45,37,44)(38,43,48,40);(1,4,6,14)(2,9,8,12)(3,11,10,16)(5,15,7,13)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,37,39,35)(34,43,41,40)(36,44,47,45)(38,42,48,46);(1,5,6,7)(2,10,8,3)(4,13,14,15)(9,11,12,16)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,38,39,48)(34,44,41,45)(35,42,37,46)(36,40,47,43)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=h,f^2=gi,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, 8632008);