#4598463

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1852 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)(9,12)(13,14,18,16)(15,21,23,19)(17,24,26,20)(22,27,25,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(5,6,8,10)(7,9,11,12)(14,16)(19,21)(20,24)(27,28)(29,30,33,36)(31,34,39,37)(32,35,41,40)(38,42,44,43);(5,6)(7,9)(8,10)(11,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(1,2)(3,4)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,7)(6,9)(8,11)(10,12)(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=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, 4505659);