#3102203

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1728 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)(3,7,8,5)(10,12)(14,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,29)(28,35)(30,31)(32,38)(33,39)(34,37)(36,40);(1,2,4,6)(3,5,8,7)(11,15)(14,16)(17,19,22,21)(18,23,20,24)(26,31)(29,39)(32,35)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,14,15,16)(17,19,22,21)(18,24,20,23)(25,27,30,33)(26,29,31,39)(28,36,38,34)(32,37,35,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,14,12,16)(17,20,22,18)(19,23,21,24)(25,28,30,38)(26,32,31,35)(27,34,33,36)(29,40,39,37);(1,3)(2,5)(4,8)(6,7)(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,23,20,24)(25,29)(26,33)(27,31)(28,37)(30,39)(32,34)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fgh,d^2=fgh,e^2=gh,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, 3009399);