#124770

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1466
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

301 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,d^b=dg,e^b=efg,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31966);