#3378904

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1847 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)(19,21,23,24)(25,26)(27,29)(28,35)(30,31)(32,36)(33,39)(34,37)(38,40);(2,3)(6,7)(11,16)(13,15)(17,19,20,23)(18,21,22,24)(26,31)(27,33)(28,36)(37,38);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,33)(26,29,31,39)(28,34,36,40)(32,38,35,37);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,35)(33,40)(37,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,27)(28,37)(30,39)(31,33)(32,40)(34,35)(36,38)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=h,d^2=fh,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, 3286100);