#10521329

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

4192 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=hi,e^2=i,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, 10428525);