#4121538

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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

2166 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fh,d^2=fh,e^2=fghi,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, 4028734);