#7715516

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	#37559	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4782 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,29,39)(20,25,33,31)(21,26,36,30)(22,37,38,27)(32,46,43,40)(34,45,47,41)(35,44,48,42);(9,11,13,12)(10,14,16,15)(19,29)(20,33)(24,39)(25,31)(32,43)(34,47)(40,46)(41,45);(2,5)(6,7)(9,10,13,16)(11,14,12,15)(17,19)(18,24)(20,30)(21,25)(22,32)(23,29)(26,33)(27,40)(28,39)(31,36)(34,44)(35,41)(37,46)(38,43)(42,47)(45,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,25)(19,30)(21,24)(22,34)(23,33)(26,29)(27,41)(28,31)(32,44)(35,40)(36,39)(37,45)(38,47)(42,43)(46,48);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,15,16,14)(17,21)(18,26)(19,31)(20,24)(22,35)(23,36)(25,29)(27,42)(28,30)(32,45)(33,39)(34,40)(37,44)(38,48)(41,43)(46,47);(17,22,23,38)(18,27,28,37)(19,32,29,43)(20,34,33,47)(21,35,36,48)(24,40,39,46)(25,41,31,45)(26,42,30,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622712);