#6629425

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#3035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2332 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,c^b=ci,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536621);