#124547

order	512
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

376 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 86 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)(9,10,12,17,13,18,26,35,14,19,25,34,28,16,31,32)(11,20,22,33,23,30,38,39,21,36,37,40,27,15,24,29);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,15)(12,24,25,38)(13,27,28,23)(16,20)(17,33)(18,36)(19,30)(22,26,37,31)(29,35)(32,39)(34,40);(12,25)(17,34)(22,37)(24,38)(26,31)(29,39)(32,35)(33,40)

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292819, #72292832, #72292835
maximal normal subgroup of	#72292819, #72292832, #72292835
maximal quotient of	#72293306

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31743);