#6542574

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×#3540	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,d^c=di,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, 6449770);