#130117

order	512
nilpotency class	4
isoclinism	isoclinic class of #130114
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 80 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)(11,19)(12,23)(13,18,28,32)(14,29)(15,20)(16,22,33,26)(17,21)(24,31,27,40)(25,36)(30,35)(34,38,39,37);(1,2)(3,7)(4,6)(5,8)(9,11,15,14)(10,16,20,33)(12,21,25,30)(13,26,28,22)(17,31,35,40)(18,29,32,19)(23,39,36,34)(24,38,27,37);(9,12,15,25)(10,17,20,35)(11,21,14,30)(13,27,28,24)(16,31,33,40)(18,34,32,39)(19,36,29,23)(22,38,26,37)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284391, #72285658
maximal normal subgroup of	#72284391, #72285658

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37313);