#348833

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

969 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 969 conjugacy classes of subgroups — too many to draw. Only its 176 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256029);