#6606624

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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)(13,14,18,22,19,23,17,30)(15,24,26,35,27,34,32,20)(16,21,28,38,29,37,31,36)(25,39,42,43,41,44,40,33);(5,6,8,9)(7,10,11,12)(13,15,19,27)(14,20,23,35)(16,25,29,41)(17,26,18,32)(21,33,37,43)(22,34,30,24)(28,40,31,42)(36,39,38,44);(1,2,3,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,31)(18,28)(19,29)(20,33)(22,38)(23,37)(24,39)(26,42)(27,41)(30,36)(32,40)(34,44)(35,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,30)(15,26,27,32)(16,28,29,31)(20,34,35,24)(21,36,37,38)(25,40,41,42)(33,39,43,44);(6,9)(7,11)(16,29)(21,37)(25,41)(28,31)(33,43)(36,38)(39,44)(40,42)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 6513820);