#7749872

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	#41943	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4369 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:

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

Polycyclic presentation:

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

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, 7657068);