#156476

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de

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, 63672);