#8661360

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

4018 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,f^2=g,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, 8568556);