#268198

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

889 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 889 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fh,e^d=ei,f^d=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, 175394);