#8922971

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

4187 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)(3,8)(4,9)(5,10)(6,7)(11,14)(12,16)(13,15)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(34,39)(35,42)(41,47)(43,44);(1,3,7,11)(2,8,6,14)(4,12,13,5)(9,16,15,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47);(2,6)(3,11)(4,13)(10,16)(19,25)(20,28)(23,27)(24,29)(35,42)(37,40)(43,44)(45,48);(1,4)(2,9)(3,12)(5,11)(6,15)(7,13)(8,16)(10,14)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,35,38,42)(34,40,39,37)(36,44,46,43)(41,45,47,48);(1,5,7,12)(2,10,6,16)(3,13,11,4)(8,15,14,9)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,36,38,46)(34,41,39,47)(35,43,42,44)(37,45,40,48);(1,6,7,2)(3,14,11,8)(4,15,13,9)(5,16,12,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,37,38,40)(34,42,39,35)(36,45,46,48)(41,44,47,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,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, 8830167);