#6606691

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

1753 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)(13,14,18,23,19,24,17,22)(15,25,27,34,28,38,32,20)(16,21,29,35,31,37,30,36)(26,39,41,44,42,43,40,33);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,26)(17,27)(18,32)(19,28)(21,33)(22,25)(23,38)(24,34)(29,40)(30,41)(31,42)(35,43)(36,39)(37,44);(1,2,3,4)(6,9)(7,11)(13,16,19,31)(14,21,24,37)(15,26,28,42)(17,30,18,29)(20,33,34,44)(22,36,23,35)(25,39,38,43)(27,41,32,40);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,29,31,30)(20,25,34,38)(21,35,37,36)(26,40,42,41)(33,43,44,39);(6,9)(7,11)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,30,31,29)(20,25,34,38)(21,36,37,35)(26,41,42,40)(33,39,44,43)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,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, 6513887);