#6596992

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6504188);