#10040709

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
prime radical	#52710	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,5)(4,10)(6,8)(7,9)(11,14)(12,15)(13,16)(17,18,23,25)(19,24,22,32)(20,28,30,21)(26,31,29,27)(35,37)(39,41)(43,44)(46,47);(1,3,7,6)(2,8,9,5)(4,12,13,11)(10,14,16,15)(17,19,23,22)(18,24,25,32)(20,29,30,26)(21,31,28,27)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(2,9)(4,13)(5,8)(11,12)(17,20,23,30)(18,21,25,28)(19,26,22,29)(24,27,32,31)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(1,4,7,13)(2,10,9,16)(3,11,6,12)(5,14,8,15)(17,20)(18,21)(19,26)(22,29)(23,30)(24,27)(25,28)(31,32)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,5,7,8)(2,3,9,6)(4,14,13,15)(10,11,16,12)(17,21)(18,20)(19,27)(22,31)(23,28)(24,26)(25,30)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,6,7,3)(2,5,9,8)(4,12,13,11)(10,14,16,15)(17,22,23,19)(18,24,25,32)(20,29,30,26)(21,27,28,31)(33,37,38,35)(34,41,42,39)(36,44,45,43)(40,47,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=h,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, 9947905);