#130133

order	512
nilpotency class	4
isoclinism	isoclinic class of #130074
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

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

normal subgroups only

This group has 567 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)(15,22,31,37)(18,29)(21,36)(23,39)(25,33,28,34)(27,35)(30,40,32,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,22)(13,27)(14,21)(16,32)(17,20)(18,30)(19,31)(23,29)(24,36)(25,40)(26,37)(28,38)(33,35)(34,39);(10,16,19,18)(11,21)(15,30,31,32)(17,34,24,33)(20,35,36,39)(22,37)(23,27)(38,40)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,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, 37329);