#8621087

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

3948 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)(3,5,8,13)(4,6,9,15)(11,12,14,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,40,43)(35,39,42,48)(36,38,37,41)(44,46,45,47);(1,3,7,8)(2,5,10,13)(4,11,9,14)(6,12,15,16)(18,24)(19,20)(26,29)(27,28)(33,35,40,42)(34,39,43,48)(36,45,37,44)(38,47,41,46);(3,8)(5,13)(11,14)(12,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,36,40,37)(34,41,43,38)(35,44,42,45)(39,47,48,46);(1,4)(2,6)(3,11)(5,12)(7,9)(8,14)(10,15)(13,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,37,40,36)(34,41,43,38)(35,45,42,44)(39,47,48,46);(1,5,7,13)(2,8,10,3)(4,12,9,16)(6,14,15,11)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,38,40,41)(34,36,43,37)(35,46,42,47)(39,44,48,45);(1,6)(2,9)(3,12)(4,10)(5,14)(7,15)(8,16)(11,13)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,39)(34,42)(35,43)(36,47)(37,46)(38,44)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8528283);