#346923

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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2007 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 2007 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)(6,11)(7,15)(8,9)(13,14)(16,17)(18,19)(22,27)(24,25)(28,35)(29,33)(30,31)(34,36);(5,6)(7,12)(8,13)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22,26,29)(23,28,32,36)(24,27,25,33)(30,35,31,34);(5,7,10,17)(6,12,14,20)(8,15,9,16)(11,18,13,19)(21,23,26,32)(22,28,29,36)(24,31,25,30)(27,34,33,35);(2,4)(5,8,10,9)(6,13,14,11)(7,16,17,15)(12,18,20,19)(21,24,26,25)(22,27,29,33)(23,30,32,31)(28,35,36,34)

Polycyclic presentation:

2^9|b^2=i,c^2=h,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, 254119);