#128361

order	512
nilpotency class	3
isoclinism	isoclinic class of #106017
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1749
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

512 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fg,c^b=ch,d^b=dg,e^b=ei,f^c=fi

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, 35557);