#5474153

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 2097152
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

1856 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)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(6,9)(10,12)(14,18)(16,24)(19,21)(22,28)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,25)(17,18)(20,24)(21,23)(22,27)(26,28)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,33,34,41)(30,36,37,44)(31,38,32,40)(35,42,39,43)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=f,e^2=fg,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, 5381349);