#122001

order	512
nilpotency class	3
isoclinism	isoclinic class of #121999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

658 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=hi,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=gi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224697, #72224703, #72224704
maximal normal subgroup of	#72224697, #72224703, #72224704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29197);