#6545452

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1593 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)(17,18,23,26,22,27,21,35)(19,28,30,39,31,24,36,40)(20,32,33,42,34,41,37,25)(29,44,47,38,46,48,45,43);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,36)(22,31)(23,30)(25,38)(26,40)(27,28)(32,48)(33,47)(34,46)(35,39)(37,45)(41,44)(42,43);(10,13)(14,16)(17,20,22,34)(18,25,27,42)(19,29,31,46)(21,33,23,37)(24,38,28,43)(26,41,35,32)(30,45,36,47)(39,48,40,44);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)(17,21,22,23)(18,26,27,35)(19,30,31,36)(20,33,34,37)(24,39,28,40)(25,41,42,32)(29,45,46,47)(38,48,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452648);