#4034378

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=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, 3941574);