#3107732

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1780 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=fhi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,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, 3014928);