#4668531

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1725 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)(6,9)(7,11)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30)(31,36)(32,39)(33,35)(34,37)(38,43)(40,44)(41,42);(6,9)(10,12)(14,15)(19,21)(20,22)(26,27)(29,31,35,32)(30,36,33,39)(34,38,42,40)(37,43,41,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,24,19)(17,20,25,22)(23,26,28,27)(29,32,35,31)(30,36,33,39)(34,40,42,38)(37,43,41,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28)(29,33,35,30)(31,36,32,39)(34,41,42,37)(38,43,40,44);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,34,35,42)(30,37,33,41)(31,38,32,40)(36,43,39,44)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=gh,d^2=hi,e^2=fg,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, 4575727);