#119739

order	512
nilpotency class	3
isoclinism	isoclinic class of #119739
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,e^c=ei,g^c=gh

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, 26935);