#4035484

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2524 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)(9,10,13,15)(11,16,14,12)(17,18)(19,25)(20,22)(21,28)(23,26)(24,29)(27,31)(30,32);(5,6,7,8)(11,14)(12,16)(18,20)(19,26)(21,29)(31,32);(5,6)(7,8)(9,11,13,14)(10,12,15,16)(17,19)(18,23)(20,25)(21,30)(22,26)(24,32)(27,29)(28,31);(1,2)(3,4)(17,20,22,18)(19,25,26,23)(21,24,29,28)(27,31,30,32);(9,12)(10,14)(11,15)(13,16)(17,21,22,29)(18,24,20,28)(19,27,26,30)(23,31,25,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fg,d^2=f,e^2=f,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, 3942680);