#8661364

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	#57125	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8568560);