#10247678

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

4016 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10154874);