#128309

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₄×#99
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

552 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=i,f^2=h,b^a=bd,c^a=ci,f^a=fg,d^b=dg,g^b=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35505);