#605220

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

1728 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,d^2=fi,e^2=fgi,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 512416);