#4435292

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

2481 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)(11,19)(12,21)(13,23)(14,15)(16,20)(17,22)(18,24)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(10,15)(11,20)(17,21)(18,23)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(5,6,8,7)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,24)(22,23)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(1,2,3,4)(5,7,8,6)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,16,24,19)(25,29,31,27)(26,33,28,32)(30,35,38,37)(34,39,36,40);(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,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, 4342488);