#119731

order	512
nilpotency class	3
isoclinism	isoclinic class of #119726
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

740 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,e^2=i,g^2=hi,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	#72213599, #72235370
maximal normal subgroup of	#72213599, #72235370
maximal quotient of	#72267649

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26927);