#8691807

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:

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

Polycyclic presentation:

2^9|a^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8599003);