#126699

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1568
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,e^c=ei

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, 33895);