#146317

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1726
prime radical	C₂⁴	#131
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 338 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)(5,6,10,17,12,20,33,46,13,18,11,19,9,16,32,45)(7,21,24,49,25,14,35,37,26,39,34,58,23,38,54,47)(8,15,28,41,30,44,56,64,31,42,29,43,27,40,55,63)(22,48,51,66,52,36,59,60,53,62,57,68,50,61,67,65);(2,4)(5,7)(6,14,18,38)(8,22)(9,23)(10,34)(11,24)(12,25)(13,26)(15,36,42,61)(16,21,20,39)(17,47,19,37)(27,50)(28,57)(29,51)(30,52)(31,53)(32,35)(33,54)(40,48,44,62)(41,65,43,60)(45,58,46,49)(55,59)(56,67)(63,68,64,66);(5,8,11,29,12,30,32,55,13,31,10,28,9,27,33,56)(6,15,19,43,20,44,45,63,18,42,17,41,16,40,46,64)(7,22,24,51,25,52,35,59,26,53,34,57,23,50,54,67)(14,36,37,60,39,62,58,68,38,61,47,65,21,48,49,66)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53513);