#8159505

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

4100 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gi,e^2=g,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^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, 8066701);