#10504584

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

4084 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=gi,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, 10411780);