#132034

order	512
nilpotency class	4
isoclinism	isoclinic class of #132019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2316 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

510 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39230);