#8701422

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(33,34,38,41)(35,39,43,48)(36,44,37,40)(42,47,45,46);(2,7)(3,10)(9,16)(11,13)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,35)(34,39)(36,45)(37,42)(38,43)(40,47)(41,48)(44,46);(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36,38,37)(34,40,41,44)(35,42,43,45)(39,46,48,47);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,15)(14,16)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,35)(34,39)(36,42)(37,45)(38,43)(40,46)(41,48)(44,47);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,13,16,11)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,34)(35,39)(36,40)(37,44)(38,41)(42,46)(43,48)(45,47);(1,5,6,15)(2,9,7,16)(3,11,10,13)(4,12,8,14)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,37,38,36)(34,40,41,44)(35,42,43,45)(39,47,48,46)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608618);