#348262

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

829 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 829 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,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, 255458);