#4063000

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=h,d^2=fh,e^2=fgh,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, 3970196);