#152114

order	512
nilpotency class	5
isoclinism	isoclinic class of #149797
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊Q₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

642 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 78 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)(5,6,9,15,11,17,29,44)(7,18,20,42,22,35,49,61)(8,13,25,38,24,39,53,66)(10,16,31,43,32,14,28,41)(12,27,34,30,36,48,62,21)(19,45,46,64,23,50,51,67)(26,37,57,63,54,40,56,65)(33,58,59,52,60,55,68,47);(5,7,11,22)(6,12,17,36)(8,23,24,19)(9,27,29,48)(10,30,32,21)(13,33,39,60)(14,35,16,18)(15,42,44,61)(20,28,49,31)(25,55,53,58)(26,47,54,52)(34,43,62,41)(37,45,40,50)(38,64,66,67)(46,57,51,56)(59,63,68,65);(2,4)(5,8)(6,13)(7,19)(9,25)(10,26)(11,24)(12,33)(14,40)(15,38)(16,37)(17,39)(18,45)(20,46)(21,47)(22,23)(27,58)(28,56)(29,53)(30,52)(31,57)(32,54)(34,59)(35,50)(36,60)(41,65)(42,64)(43,63)(44,66)(48,55)(49,51)(61,67)(62,68)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,f^b=fgh,g^b=gi,d^c=di,f^d=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59310);