#363997

order	512
nilpotency class	3
isoclinism	isoclinic class of #363903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2297
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13957
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

991 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)(3,15,19,38)(4,20,24,39)(5,25,28,36)(6,12)(7,13)(8,14)(9,18,33,29)(10,23,35,30)(11,26,37,27)(16,31)(17,45)(21,49)(22,51)(32,47)(34,53)(40,43)(41,54)(42,60,50,64)(44,57)(46,58)(48,55)(52,59)(56,61,63,62);(1,3,7,29,8,19,6,18)(2,9,12,15,14,33,13,38)(4,21,30,43,24,16,23,44)(5,17,27,54,28,47,26,46)(10,31,20,40,35,49,39,57)(11,32,25,41,37,45,36,58)(22,50,55,61,53,42,52,62)(34,56,48,60,51,63,59,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31)(11,34)(12,20)(13,39)(14,35)(15,40)(17,42)(18,43)(19,21)(25,48)(26,52)(27,55)(28,53)(29,44)(32,56)(33,49)(36,59)(37,51)(38,57)(41,60)(45,63)(46,61)(47,50)(54,62)(58,64);(1,5,8,28)(2,11)(3,17,19,47)(4,22,24,53)(6,26,7,27)(9,32)(10,34)(12,25)(13,36)(14,37)(15,41)(16,42,21,50)(18,46,29,54)(20,48)(23,52,30,55)(31,56)(33,45)(35,51)(38,58)(39,59)(40,60)(43,61,44,62)(49,63)(57,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285536, #72285922, #72285996, …
maximal normal subgroup of	#72285536, #72285922, #72285996, …
maximal quotient of	C₂×#363997

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271193);