#6606375

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 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)(13,14,19,24)(15,25,28,34)(16,21,31,38)(17,22,18,23)(20,32,35,27)(26,39,42,44)(29,36,30,37)(33,40,43,41);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,26)(17,27)(18,32)(19,28)(21,33)(22,34)(23,25)(24,35)(29,40)(30,41)(31,42)(36,39)(37,44)(38,43);(1,2,3,4)(6,9)(7,11)(13,16)(14,21)(15,26)(17,30)(18,29)(19,31)(20,33)(22,37)(23,36)(24,38)(25,39)(27,41)(28,42)(32,40)(34,44)(35,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,29,31,30)(20,34,35,25)(21,36,38,37)(26,40,42,41)(33,39,43,44);(6,9)(7,11)(13,17,19,18)(14,23,24,22)(15,27,28,32)(16,30,31,29)(20,25,35,34)(21,36,38,37)(26,41,42,40)(33,39,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,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, 6513571);