#387200

order	512
nilpotency class	3
isoclinism	isoclinic class of #232138
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 226 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:

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

Polycyclic presentation:

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

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, 294396);