#496065

order	512
nilpotency class	4
isoclinism	isoclinic class of #483956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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)(9,10,14,19,16,20,15,22)(11,21,23,12,17,24,18,13);(2,5)(6,8)(9,11,13,20,15,23,24,22,16,17,12,10,14,18,21,19);(1,3,4,7)(2,6,5,8)(9,11,13,20,15,23,24,22,16,17,12,10,14,18,21,19);(9,12,14,24,16,13,15,21)(10,17,19,18,20,11,22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225296, #72225297, #72225298, …
maximal normal subgroup of	#72225296, #72225297, #72225298, …
maximal quotient of	#72343508, #72343603

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403261);