#8701363

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gi,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, 8608559);