#6621078

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1726 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)(13,14)(15,25)(16,30)(17,24)(18,23)(19,22)(20,28)(21,32)(26,40)(27,36)(29,35)(31,37)(33,38)(34,41)(39,43)(42,44);(1,2,3,4)(6,10)(8,11)(13,15,19,27)(14,20,22,35)(16,26,33,41)(17,29,18,28)(21,34,37,40)(23,36,24,25)(30,44,38,39)(31,43,32,42);(5,6)(7,8)(9,10)(11,12)(13,16,19,33)(14,21,22,37)(15,26,27,41)(17,32,18,31)(20,34,35,40)(23,30,24,38)(25,39,36,44)(28,43,29,42);(5,7,9,12)(6,11,10,8)(15,27)(20,35)(25,36)(26,41)(28,29)(34,40)(39,44)(42,43);(5,8,9,11)(6,7,10,12)(14,22)(20,35)(21,37)(23,24)(25,36)(30,38)(34,40)(39,44)

Polycyclic presentation:

2^9|b^2=fgi,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6528274);