#582335

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1748 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)(7,11)(10,12)(14,18)(19,24)(20,25)(26,28)(29,30)(31,37)(32,40)(33,34)(35,39)(36,41)(38,42)(43,44);(6,9)(10,12)(14,18)(15,21)(20,25)(22,27)(30,34)(35,37)(36,40)(42,43);(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,30)(31,37)(32,36)(33,34)(35,39)(38,43)(40,41)(42,44);(5,7,8,11)(6,10,9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31,33,39)(30,35,34,37)(32,38,41,44)(36,42,40,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 489531);