#583269

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fhi,d^2=fh,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, 490465);