#382950

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

602 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290146);