#138397

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	#32913	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

569 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gh,d^2=ghi,e^2=i,f^2=ghi,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=fhi,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, 45593);