#8662902

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

4013 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,22,26)(19,23,20,24)(21,25,30,29)(27,31,28,32)(34,39)(36,45)(40,46)(43,48);(2,7)(3,10)(9,12)(11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(35,42)(40,46)(43,48)(44,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,34,38,39)(35,40,42,46)(36,41,45,37)(43,47,48,44);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,35)(34,40)(36,43)(37,44)(38,42)(39,46)(41,47)(45,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,36)(34,41)(35,43)(37,39)(38,45)(40,47)(42,48)(44,46);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,38,41)(34,36,39,45)(35,44,42,47)(40,43,46,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570098);