#4381114

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=g,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4288310);