#8661372

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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:

(1,2,7,5)(3,10,8,4)(6,9,15,13)(11,14,16,12)(18,21)(19,20)(26,29)(27,28)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,14)(15,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(35,42)(39,46)(43,47)(44,48);(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,4,7,10)(2,8,5,3)(6,14,15,12)(9,11,13,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5,7,2)(3,10,8,4)(6,9,15,13)(11,12,16,14)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,6,7,15)(2,9,5,13)(3,11,8,16)(4,12,10,14)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(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,d^2=g,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, 8568568);