#8139504

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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)(3,10,6,8)(4,12,14,9)(11,15,13,16)(17,18,21,25)(19,23,26,22)(20,28,29,24)(27,30,32,31)(34,40)(35,36)(37,47)(38,48);(1,3,7,6)(2,8,5,10)(4,11,14,13)(9,15,12,16)(19,26)(22,23)(27,32)(30,31)(33,34)(35,41)(36,45)(37,42)(38,43)(39,40)(44,47)(46,48);(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35,39,36)(34,41,40,45)(37,43,47,46)(38,44,48,42);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36,39,35)(34,41,40,45)(37,43,47,46)(38,42,48,44);(1,5,7,2)(3,10,6,8)(4,9,14,12)(11,15,13,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,37,39,47)(34,42,40,44)(35,43,36,46)(38,45,48,41);(1,6,7,3)(2,10,5,8)(4,13,14,11)(9,16,12,15)(17,20,21,29)(18,24,25,28)(19,27,26,32)(22,30,23,31)(33,38)(34,43)(35,44)(36,42)(37,45)(39,48)(40,46)(41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8046700);