#3531593

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1571 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=fgi,e^2=f,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, 3438789);