#4033900

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,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, 3941096);