#8700321

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

3999 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=g,e^2=gh,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, 8607517);