#719704

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

3294 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)(11,15)(12,13)(14,16)(17,18,20,21)(19,22,24,23);(6,8)(10,13)(14,15)(17,18)(19,23)(20,21)(22,24);(5,6,7,8)(11,16)(14,15)(18,21)(22,23);(1,2,3,4)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24);(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 626900);