#4121541

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 2097152
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=fh,d^2=h,e^2=ghi,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, 4028737);