#348312

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1015 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255508);