#606536

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

1892 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)(5,6,10,12)(7,9,8,11)(14,18)(15,21)(16,23)(26,28)(29,30,35,33)(31,37,32,36)(34,41,42,38)(39,43,40,44);(5,7)(6,11)(8,10)(9,12)(14,18)(15,21)(20,25)(22,27)(29,31,35,32)(30,36,33,37)(34,40,42,39)(38,44,41,43);(1,2,4,3)(6,12)(7,8)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,32,35,31)(30,37,33,36)(34,40,42,39)(38,44,41,43);(1,2,4,3)(5,8,10,7)(6,9,12,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,33,35,30)(31,36,32,37)(34,41,42,38)(39,43,40,44);(1,3,4,2)(5,9)(6,7)(8,12)(10,11)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,34,35,42)(30,38,33,41)(31,39,32,40)(36,43,37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=i,e^2=fghi,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, 513732);