#129840

order	512
nilpotency class	4
isoclinism	isoclinic class of #129837
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 ◁ #2274 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

554 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37036);