#479837

order	512
nilpotency class	3
isoclinism	isoclinic class of #478195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 932 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)(9,10,14,19,16,20,15,11)(12,21,24,17,23,18,13,22);(2,5)(3,7)(9,11,15,20,16,19,14,10)(12,17,24,18,23,22,13,21);(3,7)(6,8)(9,12,14,24,16,23,15,13)(10,17,11,21,20,22,19,18);(1,3,4,7)(2,6,5,8)(9,13,14,23,16,24,15,12)(10,18,11,22,20,21,19,17)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387033);