#8634380

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

3975 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=g,f^2=gi,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, 8541576);