#479836

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

831 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 831 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227424
maximal normal subgroup of	#72227424
maximal quotient of	#72268633

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387032);