#8127483

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	#59758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034679);