#132105

order	512
nilpotency class	4
isoclinism	isoclinic class of #131985
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	#35463	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1120 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,6,8,5)(4,7)(9,10,14,19,15,20,13,18)(11,21,25,16,24,31,23,30)(12,17,26,32,27,33,28,34)(22,36,37,40,38,35,39,29);(2,5)(3,8)(6,7)(9,11)(10,16)(12,22)(13,25)(14,23)(15,24)(17,29)(18,31)(19,21)(20,30)(26,39)(27,38)(28,37)(32,35)(33,40)(34,36);(9,12)(10,17)(11,22)(13,28)(14,26)(15,27)(16,29)(18,34)(19,32)(20,33)(21,35)(23,39)(24,38)(25,37)(30,40)(31,36)

Polycyclic presentation:

2^9|a^2=f,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266435, #72268762
maximal normal subgroup of	#72268762
maximal quotient of	#72266414

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39301);