#7658868

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	#36670	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566064);