#134012

order	512
nilpotency class	4
isoclinism	isoclinic class of #133877
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

1112 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1112 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)(3,7,8,5)(4,6)(9,10)(11,19,22,34)(12,16)(13,26,27,17)(14,18)(15,21)(20,33,23,37)(24,39,40,31)(25,32)(28,35)(29,36)(30,38);(9,11)(10,15,18,29)(12,23)(13,21)(14,22)(16,30,32,28)(17,19,26,34)(20,25)(24,38)(27,36)(31,37,39,33)(35,40);(1,3,4,8)(2,5,6,7)(9,12,14,25)(10,16,18,32)(11,20,22,23)(13,24,27,40)(15,28,29,30)(17,31,26,39)(19,33,34,37)(21,35,36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266411, #72268770, #72268771
maximal normal subgroup of	#72266411, #72268770, #72268771
maximal quotient of	#72266417

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41208);