#8810334

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

4080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gh,e^2=g,f^2=gi,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, 8717530);