#6545561

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452757);