#357971

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239933
maximal normal subgroup of	#72239933
maximal quotient of	#72239944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265167);