#152258

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

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	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

319 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,e^c=ehi,f^c=fgh,g^c=gi,f^e=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, 59454);