#6454618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3037 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)(7,10)(8,9)(11,12)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,27)(25,34)(26,38)(28,39)(29,35)(30,36)(31,37)(33,41)(40,44)(42,43);(1,2)(3,4)(13,15,18,27,19,28,16,25)(14,20,21,24,23,34,32,39)(17,26,30,41,31,42,29,40)(22,33,35,38,37,44,36,43);(2,4)(6,9)(10,12)(13,16,19,18)(14,21,23,32)(15,25,28,27)(17,30,31,29)(20,24,34,39)(22,36,37,35)(26,41,42,40)(33,43,44,38);(2,4)(13,17,19,31)(14,22,23,37)(15,26,28,42)(16,29,18,30)(20,33,34,44)(21,35,32,36)(24,38,39,43)(25,40,27,41);(5,7,8,11)(6,10,9,12)(17,31)(22,37)(26,42)(29,30)(33,44)(35,36)(38,43)(40,41)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361814);