#6606703

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1773 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)(13,14,19,32,18,23,17,22)(15,24,26,20,28,34,27,35)(16,21,29,36,31,38,30,37)(25,39,41,33,42,43,40,44);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,25)(17,26)(18,28)(19,27)(21,33)(22,34)(23,35)(24,32)(29,40)(30,41)(31,42)(36,39)(37,43)(38,44);(1,2,3,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,30)(18,31)(19,29)(20,33)(22,37)(23,38)(24,39)(26,41)(27,40)(28,42)(32,36)(34,43)(35,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,32)(15,26,28,27)(16,29,31,30)(20,34,35,24)(21,36,38,37)(25,40,42,41)(33,39,44,43);(6,9)(7,11)(13,17,18,19)(14,22,23,32)(15,26,28,27)(16,30,31,29)(20,34,35,24)(21,37,38,36)(25,41,42,40)(33,43,44,39)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513899);