#6544915

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#39335	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2047 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)(15,24)(16,21)(17,22)(18,33)(19,23)(20,26)(25,41)(27,35)(28,44)(29,38)(30,39)(31,36)(32,40)(34,43)(37,42);(5,6)(7,10)(8,9)(11,12)(13,15,17,31,19,27,18,26)(14,20,33,35,23,36,22,24)(16,28,32,43,30,25,29,42)(21,37,38,41,39,34,40,44);(2,4)(6,9)(10,12)(13,16,19,30)(14,21,23,39)(15,25,27,28)(17,32,18,29)(20,34,36,37)(22,40,33,38)(24,41,35,44)(26,43,31,42);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,33)(15,26,27,31)(16,29,30,32)(20,35,36,24)(21,38,39,40)(25,42,28,43)(34,41,37,44);(7,11)(10,12)(15,27)(16,30)(20,36)(21,39)(24,35)(26,31)(29,32)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452111);