#127355

order	512
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	H(3,ℤ/4)
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#7980
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

336 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=ghi,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,e^b=eh,e^d=ei

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, 34551);