#648525

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

2932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,d^2=g,e^2=f,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, 555721);