#121904

order	512
nilpotency class	3
isoclinism	isoclinic class of #121899
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=dei,d^2=hi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=ei,f^c=fhi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29100);