#333515

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#1624	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

776 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 142 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)(5,6,11,14,10,16,9,15)(7,17,18,12,13,20,19,8)(21,22)(23,28)(24,29)(25,27)(26,30)(31,36)(32,34)(33,35);(5,7)(6,12)(8,16)(9,18)(10,13)(11,19)(14,17)(15,20)(22,27)(23,31)(24,26)(28,35)(32,33)(34,36);(2,4)(5,8,10,12)(6,13,16,7)(9,20,11,17)(14,19,15,18)(21,23)(22,28)(24,31)(25,32)(26,33)(27,34)(29,36)(30,35);(2,4)(6,14)(7,18)(9,11)(13,19)(15,16)(17,20)(21,24,25,26)(22,29,27,30)(23,31,32,33)(28,36,34,35)

Polycyclic presentation:

2^9|a^2=ehi,c^2=h,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240711);