#4435091

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=i,e^2=h,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, 4342287);