#6574774

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1934 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,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, 6481970);