#119895

order	512
nilpotency class	3
isoclinism	isoclinic class of #10564
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343022, #72343034, #72343038, …
maximal normal subgroup of	#72343022, #72343034, #72343038, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27091);