#8581775

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

3983 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=i,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488971);