#494888

order	512
nilpotency class	4
isoclinism	isoclinic class of #483807
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=eh,d^2=i,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343342, #72343344, #72343366, …
maximal normal subgroup of	#72343342, #72343344, #72343366, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402084);