#4419565

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4326761);