#4313367

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3752 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)(6,8)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,26,24,21)(22,28,31,25)(27,29,30,32);(10,12)(14,15)(17,19,23,24)(18,21,20,26)(22,30,31,27)(25,32,28,29);(5,6)(7,8)(11,16)(14,15)(17,20,23,18)(19,21,24,26)(22,28,31,25)(27,29,30,32);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,20)(22,29)(23,26)(25,30)(27,28)(31,32);(9,12,13,10)(11,15,16,14)(17,22,23,31)(18,25,20,28)(19,27,24,30)(21,29,26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4220563);