#4397301

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1648 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fi,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4304497);