#606841

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
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

1658 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgh,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

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, 514037);