#333540

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

716 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 146 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)(5,6,10,13,11,15,8,18)(7,16,14,9,19,20,17,12)(23,30)(27,35)(31,36)(32,34);(5,7,11,19)(6,12,15,9)(8,14,10,17)(13,20,18,16)(21,22,25,29)(23,31,30,36)(24,28,26,33)(27,34,35,32);(2,4)(5,8,11,10)(6,13,15,18)(7,17,19,14)(9,20,12,16)(21,23,25,30)(22,27,29,35)(24,32,26,34)(28,36,33,31);(2,4)(5,9,11,12)(6,14,15,17)(7,18,19,13)(8,20,10,16)(21,24,25,26)(22,28,29,33)(23,32,30,34)(27,36,35,31)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=hi,d^2=hi,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, 240736);