#126700

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

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,c^b=ci,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, 33896);