#110221

order	512
nilpotency class	3
isoclinism	isoclinic class of #110189
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

747 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 747 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,7,5,8,3,6)(9,10)(11,20,23,16)(12,24)(13,28)(14,19)(15,18)(17,26)(21,35,36,40)(22,31,29,32)(25,38)(27,33)(30,37,34,39);(2,6,8,7)(9,11,14,22,15,23,13,29)(10,16,19,32,18,20,28,31)(12,21,27,37,25,36,26,39)(17,30,24,35,33,34,38,40);(2,6,8,7)(9,12,13,26,15,25,14,27)(10,17,19,24,18,33,28,38)(11,21,29,39,23,36,22,37)(16,30,32,35,20,34,31,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257501
maximal normal subgroup of	#72257501
maximal quotient of	#72257731, #72257745

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17417);