#122128

order	512
nilpotency class	3
isoclinism	isoclinic class of #122126
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

655 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,d^c=dhi,e^c=eh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29324);