#129288

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1782
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

714 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,f^b=fg,f^c=fi,f^d=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, 36484);