#299342

order	512
nilpotency class	3
isoclinism	isoclinic class of #299341
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206538);