#10316828

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

3983 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)(3,10)(4,9)(5,14)(6,7)(8,11)(12,16)(13,15)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(36,44)(40,46)(43,45)(47,48);(2,7)(3,11)(4,14)(13,16)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(34,38)(35,41)(40,46)(43,45);(3,11)(8,10)(12,15)(13,16)(18,25)(19,28)(20,30)(26,32)(33,34,37,38)(35,39,41,42)(36,40,44,46)(43,47,45,48);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,46)(45,47);(1,4,6,14)(2,9,7,5)(3,12,11,15)(8,13,10,16)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,35)(34,39)(36,45)(37,41)(38,42)(40,48)(43,44)(46,47);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10224024);