#363957

order	512
nilpotency class	3
isoclinism	isoclinic class of #363956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13956
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2451 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)(3,8,15,29)(4,17,20,37)(5,10,21,34)(6,12)(7,11)(9,19,32,24)(13,38)(14,42)(16,28,23,36)(18,46)(22,33,49,59)(25,35)(26,45)(27,44)(30,41)(31,51)(39,61,63,60)(40,54)(43,57)(47,58)(48,56)(50,53)(52,64,55,62);(1,3)(2,8,11,29)(4,13)(5,14)(6,23)(7,15)(9,30,32,26)(10,27,34,42)(12,36,35,28)(16,25)(17,38,37,54)(18,39)(19,45)(20,40)(21,44)(22,50)(24,41)(31,55,56,52)(33,57,59,53)(43,49)(46,60,58,61)(47,63)(48,64)(51,62);(1,4)(2,9)(3,13)(5,18)(6,24)(7,20)(8,26)(10,31)(11,32)(12,37)(14,39)(15,40)(16,45)(17,35)(19,25)(21,47)(22,51)(23,41)(27,52)(28,54)(29,30)(33,58)(34,56)(36,38)(42,55)(43,64)(44,63)(46,59)(48,49)(50,62)(53,61)(57,60);(1,5,7,21)(2,10)(3,14,15,44)(4,18,20,47)(6,22,25,49)(8,27)(9,31)(11,34)(12,33)(13,39,40,63)(16,43,23,50)(17,46)(19,48,24,51)(26,52)(28,53)(29,42)(30,55)(32,56)(35,59)(36,57)(37,58)(38,60)(41,62,45,64)(54,61)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gi,f^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,h^b=hi,f^c=fi,h^c=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271153);