#122111

order	512
nilpotency class	3
isoclinism	isoclinic class of #122108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

856 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,7,8,5)(4,6)(9,10,14,20,15,19,13,18)(11,21,25,31,23,16,24,32)(12,26,28,34,27,35,29,17)(22,36,37,30,38,40,39,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16,19,21)(12,22)(13,24)(14,25)(15,23)(17,33,34,30)(18,31,20,32)(26,36,35,40)(27,38)(28,37)(29,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17,19,34)(11,22)(13,28)(14,29)(15,27)(16,30,21,33)(18,26,20,35)(23,38)(24,37)(25,39)(31,40,32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255848, #72255866, #72255873, …
maximal normal subgroup of	#72255848, #72255866, #72255873, …
maximal quotient of	#72266274

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29307);