#606111

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

1782 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)(6,7)(8,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,28)(24,25)(26,27)(30,31)(33,41)(35,37)(39,44);(6,7)(10,11)(14,19)(17,20)(21,27)(25,28)(29,30,34,31)(32,37,40,35)(33,38,41,36)(39,42,44,43);(1,2,4,3)(5,6,9,7)(8,10,12,11)(13,15,18,22)(14,20,19,17)(16,24,26,23)(21,25,27,28)(29,31,34,30)(32,35,40,37)(33,38,41,36)(39,42,44,43);(1,2,4,3)(5,7,9,6)(8,11,12,10)(13,16,18,26)(14,21,19,27)(15,23,22,24)(17,25,20,28)(29,32,34,40)(30,35,31,37)(33,39,41,44)(36,42,38,43);(1,3,4,2)(5,8,9,12)(6,10,7,11)(13,17,18,20)(14,22,19,15)(16,25,26,28)(21,24,27,23)(29,33)(30,36)(31,38)(32,39)(34,41)(35,42)(37,43)(40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fhi,d^2=fhi,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, 513307);