#138034

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

586 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,7)(6,8)(9,10,14,17,15,20,13,19)(11,21,26,38,23,39,25,37)(12,27,29,32,30,18,22,35)(16,31,33,28,34,40,36,24);(1,2,3,5,4,6,7,8)(9,11,22,40)(10,16,32,37)(12,28,13,26)(14,25,30,24)(15,23,29,31)(17,33,27,39)(18,21,19,36)(20,34,35,38);(2,6)(5,8)(10,17)(11,23)(13,14)(16,33)(18,32)(19,20)(21,37)(22,29)(24,28)(27,35)(34,36)(38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=gh,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 45230);