#4084644

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×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

1601 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fhi,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,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, 3991840);