#185825

order	512
nilpotency class	3
isoclinism	isoclinic class of #185503
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,d^c=di,g^c=gh

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, 93021);