#318611

order	512
nilpotency class	3
isoclinism	isoclinic class of #318601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19280	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 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)(3,6)(7,8)(9,10)(11,21)(12,16)(13,20)(14,19)(15,18)(17,23)(22,24);(1,2,3,5,4,7,6,8)(9,11,13,23,14,12,15,22)(10,16,20,24,19,21,18,17);(9,12,14,11)(10,17,19,24)(13,23,15,22)(16,20,21,18);(1,2,6,8,4,7,3,5)(9,13,14,15)(10,18,19,20)(11,22,12,23)(16,24,21,17)

Polycyclic presentation:

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

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, 225807);