#368867

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14023
prime radical	#60422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342961, #72342962, #72342979
maximal normal subgroup of	#72342961, #72342962, #72342979
maximal quotient of	#72343034

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276063);