#185746

order	512
nilpotency class	3
isoclinism	isoclinic class of #185463
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×#3511	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2370 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92942);