#110275

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#127
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

891 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 182 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 — 3 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17471);