#186307

order	512
nilpotency class	3
isoclinism	isoclinic class of #186146
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2339 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239884
maximal normal subgroup of	#72239884
maximal quotient of	#72239896

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93503);