#10521334

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

3976 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=hi,e^2=gi,f^2=g,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, 10428530);