#149181

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1843
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 206 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=i,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,e^b=ei,d^c=di

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, 56377);