#352686

order	512
nilpotency class	3
isoclinism	isoclinic class of #352102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 194 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)(10,16)(11,12)(13,14)(17,18)(19,27)(20,23)(21,31)(22,24)(25,29)(26,28)(30,32);(2,5)(6,8)(9,10,13,16,15,12,14,11)(17,19,28,30,22,29,21,20)(18,23,31,25,24,32,26,27);(9,11,13,10,15,16,14,12)(18,24)(19,29)(20,30)(26,31);(1,3,4,7)(2,6,5,8)(9,12,13,11,15,10,14,16)(17,20,21,29,22,30,28,19)(18,25,26,23,24,27,31,32)

Polycyclic presentation:

2^9|b^2=ghi,c^2=fh,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^c=dh

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, 259882);