#5474785

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1977 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)(5,6,8,10)(7,9,11,12)(14,18)(15,21)(20,25)(23,27)(29,30,34,37)(31,38,35,32)(33,36,42,44)(39,43,41,40);(7,11)(9,12)(16,24)(20,25)(23,27)(26,28)(29,31,34,35)(30,32,37,38)(33,39,42,41)(36,40,44,43);(5,7)(6,9)(8,11)(10,12)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31)(30,32)(33,41)(34,35)(36,43)(37,38)(39,42)(40,44);(1,2)(3,4)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32)(30,35)(31,37)(33,40)(34,38)(36,41)(39,44)(42,43);(5,6)(7,9)(8,10)(11,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,41)(37,44)(38,43)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=fg,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, 5381981);