#8772816

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

4219 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8680012);