#391974

order	512
nilpotency class	3
isoclinism	isoclinic class of #232147
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

844 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,15,20,13,18,14,19)(11,16,22,31,24,33,23,32)(12,25,29,17,27,36,28,35)(21,37,26,30,39,34,38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16,18,33)(12,26,27,38)(13,24)(14,23)(15,22)(17,34)(19,32,20,31)(21,29,39,28)(25,30)(35,37)(36,40);(2,6)(5,8)(9,12)(10,17)(11,21)(13,27)(14,29)(15,28)(16,30)(18,35)(19,36)(20,25)(22,38)(23,26)(24,39)(31,37)(32,34)(33,40);(3,7)(5,8)(11,22,24,23)(12,27)(16,31,33,32)(17,35)(21,38,39,26)(25,36)(28,29)(30,37,40,34)

Polycyclic presentation:

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

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, 299170);