#8533081

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

3891 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440277);