#5509591

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1735 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5416787);