#102143

order	512
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#390
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

482 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=e,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=chi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210672
maximal normal subgroup of	#72210672
maximal quotient of	#72210504, #72210557

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9339);