#8158753

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	#53053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(3,9)(6,12)(10,15)(13,16)(18,24)(20,25)(26,30)(28,32)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29)(33,35,38,46)(34,39,42,43)(36,44,37,45)(40,47,41,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,21,23,19)(18,20,24,25)(22,29,31,27)(26,28,30,32)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,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,d^c=dgi,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, 8065949);