#346929

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3548	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2356 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 2356 conjugacy classes of subgroups — too many to draw. Only its 200 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

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, 254125);