#93343

order	512
nilpotency class	4
isoclinism	isoclinic class of #7213
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#342
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×#405	C₂²
socle	C₂²	#1282
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,5)(4,7)(6,8)(9,10,13,19,15,22,35,46,16,21,14,20,34,45,36,47)(11,23,25,48,27,51,56,70,28,50,26,49,55,69,57,71)(12,29,30,44,31,58,59,68,32,18,33,41,60,42,61,66)(17,37,38,62,40,63,64,24,39,52,43,53,65,72,67,54);(2,5,7,6)(9,11,14,26,15,27,36,57,16,28,13,25,34,55,35,56)(10,17,20,38,22,40,47,64,21,39,19,43,45,65,46,67)(12,24,30,52,31,53,59,72,32,54,33,37,60,62,61,63)(18,23,41,49,42,51,66,71,29,50,44,48,58,69,68,70)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,g^a=gi,d^b=dg,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 539);