#10464588

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

4113 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)(3,10)(4,7)(5,8)(6,9)(11,15)(12,13)(14,16)(18,22)(20,28)(23,25)(26,29)(33,34,38,41)(35,39,44,37)(36,40,46,48)(42,47,43,45);(1,3,6,8)(2,5,9,10)(4,12,14,11)(7,15,16,13)(18,22)(23,25)(24,30)(31,32)(33,35,38,44)(34,39,41,37)(36,43,46,42)(40,45,48,47);(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,4,6,14)(2,7,9,16)(3,11,8,12)(5,13,10,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,34)(35,37)(36,40)(38,41)(39,44)(42,45)(43,47)(46,48);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,15)(14,16)(17,19,21,27)(18,23,22,25)(20,29,28,26)(24,32,30,31)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,5)(2,8)(3,9)(4,13)(6,10)(7,12)(11,16)(14,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,37)(34,35)(36,45)(38,39)(40,42)(41,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371784);