#605601

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

1732 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)(15,20)(16,22)(17,21)(18,19)(23,27)(24,28)(25,26)(30,34)(31,37)(32,40)(42,44);(6,9)(10,12)(14,19)(15,22)(17,25)(24,27)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,20,19,16)(17,23,25,28)(21,27,26,24)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16,18,20)(14,15,19,22)(17,24,25,27)(21,23,26,28)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,43)(41,44);(1,3,4,2)(13,17,18,25)(14,21,19,26)(15,23,22,28)(16,24,20,27)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgi,d^2=ghi,e^2=fi,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, 512797);