#10218592

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
prime radical	#47828	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4502 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,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, 10125788);