#6552823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1805 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6460019);