#4397049

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

1842 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)(3,8)(4,7)(5,6)(9,10,14,12)(11,16,13,15)(17,18)(19,20)(21,22)(23,24)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39);(2,6)(3,7)(9,11,14,13)(10,15,12,16)(18,22)(20,23)(26,27)(28,35)(32,34)(36,40);(2,6)(4,8)(9,10,14,12)(11,15,13,16)(17,19,21,24)(18,23,22,20)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,3,5,7)(2,4,6,8)(9,12,14,10)(11,15,13,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,13,14,11)(10,15,12,16)(17,19)(18,20)(21,24)(22,23)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fhi,d^2=fgi,e^2=fg,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, 4304245);