#119730

order	512
nilpotency class	3
isoclinism	isoclinic class of #119726
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267641
maximal normal subgroup of	#72267641
maximal quotient of	#72267649

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26926);