#648694

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(2,4)(9,10,13,15)(11,12,16,14)(17,18,22,21)(19,24,23,20);(5,6,7,8)(9,11,13,16)(10,14,15,12);(5,6)(7,8)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4)(9,12)(10,11)(13,14)(15,16)(17,20,22,24)(18,23,21,19);(9,11)(10,12)(13,16)(14,15)(17,21,22,18)(19,24,23,20)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 555890);