#119258

order	512
nilpotency class	3
isoclinism	isoclinic class of #119249
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
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

636 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=d,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,e^c=eh,g^c=gh

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, 26454);