#6542379

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#3547	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2989 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)(13,14,19,22,18,23,17,24)(15,25,27,20,29,38,28,35)(16,30,31,36,32,21,33,37)(26,40,43,44,42,39,41,34);(5,6)(7,10)(8,9)(11,12)(13,15,18,29)(14,20,23,35)(16,26,32,42)(17,27,19,28)(21,34,30,44)(22,25,24,38)(31,41,33,43)(36,40,37,39);(2,4)(6,9)(10,12)(13,16,18,32)(14,21,23,30)(15,26,29,42)(17,33,19,31)(20,34,35,44)(22,37,24,36)(25,39,38,40)(27,43,28,41);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,24)(15,27,29,28)(16,31,32,33)(20,25,35,38)(21,36,30,37)(26,41,42,43)(34,40,44,39);(7,11)(10,12)(14,23)(16,32)(20,35)(22,24)(25,38)(26,42)(31,33)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449575);