#391880

order	512
nilpotency class	3
isoclinism	isoclinic class of #391879
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14662
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,f^c=fi,h^d=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264914, #72264989
maximal normal subgroup of	#72264914, #72264989
maximal quotient of	#72265565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299076);