#391656

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 190 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)(3,6)(4,5)(7,8)(9,10,15,21,16,20,14,33)(11,22,26,40,27,17,25,36)(12,28,30,18,13,19,32,38)(23,35,29,39,24,37,31,34);(3,7)(6,8)(9,11,16,27)(10,17,20,22)(12,29,13,31)(14,26,15,25)(18,37,38,35)(19,39,28,34)(21,40,33,36)(23,32,24,30);(2,5)(6,8)(9,12,16,13)(10,18,20,38)(11,23,27,24)(14,32,15,30)(17,34,22,39)(19,33,28,21)(25,31,26,29)(35,40,37,36);(1,3,4,7)(2,6,5,8)(9,13,16,12)(10,19,20,28)(11,24,27,23)(14,30,15,32)(17,35,22,37)(18,21,38,33)(25,29,26,31)(34,36,39,40)

Polycyclic presentation:

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

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, 298852);