#197374

order	512
nilpotency class	3
isoclinism	isoclinic class of #197321
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35237	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1457 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104570);