#10020623

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

4301 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9927819);