#8115569

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022765);