#184945

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#3460	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2466 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,e^b=eh,f^b=fh,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92141);