#8688329

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595525);