#134038

order	512
nilpotency class	4
isoclinism	isoclinic class of #133825
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1427 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1427 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,f^c=fh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268797, #72268802
maximal normal subgroup of	#72268797, #72268802
maximal quotient of	#72268853

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41234);