#94062

order	512
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂³	C₄×C₁₆
Frattini, cosocle	C₂³×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,17,13,19,26,37,14,20,27,38,29,41,50,61,15,18,28,36,30,39,51,59,31,40,52,60,53,62,68,71)(11,21,23,42,24,43,47,63,25,44,48,64,49,65,67,72,22,16,32,33,45,34,54,56,46,35,55,57,66,58,69,70);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16,18,21)(12,23)(13,24)(14,25)(15,22)(17,33,36,42)(19,34,39,43)(20,35,40,44)(26,47)(27,48)(28,32)(29,49)(30,45)(31,46)(37,56,59,63)(38,57,60,64)(41,58,62,65)(50,67)(51,54)(52,55)(53,66)(61,70,71,72)(68,69)

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ei,c^b=ci,d^b=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1258);