#137774

order	512
nilpotency class	4
isoclinism	isoclinic class of #137768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

651 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 651 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dgi,e^b=ei,f^b=fgi,g^b=gi,d^c=di,e^c=ei,f^c=fi,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, 44970);