#4423097

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2372 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fh,c^2=fg,d^2=g,e^2=fh,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, 4330293);