#10519395

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

4167 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)(3,10)(4,7)(5,8)(6,9)(11,15)(12,13)(14,16)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34)(35,42)(36,40)(37,45)(38,39)(41,46)(43,47)(44,48);(1,3,6,8)(2,5,9,10)(4,12,14,11)(7,15,16,13)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(34,39)(40,42)(41,45)(47,48);(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,35,38,36)(34,40,39,42)(37,43,46,44)(41,47,45,48);(1,4,6,14)(2,7,9,16)(3,11,8,12)(5,13,10,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,34)(35,40)(36,42)(37,41)(38,39)(43,47)(44,48)(45,46);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,15)(14,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36,38,35)(34,40,39,42)(37,43,46,44)(41,48,45,47);(1,5)(2,8)(3,9)(4,13)(6,10)(7,12)(11,16)(14,15)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,37,38,46)(34,41,39,45)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=ghi,e^2=hi,f^2=h,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, 10426591);