#352807

order	512
nilpotency class	3
isoclinism	isoclinic class of #352174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fhi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=di,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260003);