#3333304

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2178 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)(9,10,13,14)(11,17,19,15)(12,20,22,16)(18,23,21,24)(25,26,30,28)(27,32,35,31)(29,33,38,36)(34,40,37,39);(6,8)(10,14)(15,17)(16,20)(23,24)(25,27,30,35)(26,31,28,32)(29,34,38,37)(33,39,36,40);(5,6)(7,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(1,2)(3,4)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,28,30,26)(27,31,35,32)(29,33,38,36)(34,40,37,39);(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3240500);