#130116

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 662 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)(11,20)(12,24,26,17)(13,16)(14,19)(15,21,32,39)(18,28)(22,35)(23,37)(25,33,29,34)(27,36)(30,38,31,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,23)(10,15,19,32)(12,21,26,39)(13,27,28,22)(16,30,18,31)(17,20,24,37)(25,40,29,38)(33,36,34,35);(10,16)(13,28)(15,30)(17,34)(18,19)(20,35)(22,27)(24,33)(25,29)(31,32)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37312);