#8634356

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	#2416	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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,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, 8541552);