#4664750

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

1779 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)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30)(31,38)(32,36)(33,43)(34,44)(35,37)(39,40)(41,42);(6,9)(10,12)(15,22)(18,21)(23,28)(25,27)(29,31,35,32)(30,36,37,38)(33,39,44,41)(34,40,43,42);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,32,35,31)(30,36,37,38)(33,41,44,39)(34,40,43,42);(1,2,3,4)(13,15,17,22)(14,18,20,21)(16,23,24,28)(19,25,26,27)(29,33)(30,34)(31,39)(32,41)(35,44)(36,40)(37,43)(38,42);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,34)(30,33)(31,40)(32,42)(35,43)(36,39)(37,44)(38,41)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=fi,e^2=g,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, 4571946);