#8165356

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	#59674	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4012 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=h,e^2=i,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, 8072552);