#395175

order	512
nilpotency class	3
isoclinism	isoclinic class of #395172
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14722
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1693 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)(4,20)(5,24)(6,13)(7,12)(8,14)(9,18)(10,22)(11,27)(16,30)(17,46)(19,38)(21,51)(23,39)(25,36)(26,37)(28,32)(29,35)(31,56)(33,45)(34,57)(40,44)(41,48)(42,59)(43,49)(47,54)(50,53)(52,55)(58,63)(60,61)(62,64);(1,3,6,18,8,19,7,28)(2,9,12,15,14,32,13,38)(4,16,22,43,23,45,29,44)(5,17,25,47,27,48,26,56)(10,33,20,49,35,30,39,40)(11,31,36,41,24,54,37,46)(21,42,52,61,53,63,57,62)(34,59,50,64,55,58,51,60);(1,4)(2,10)(3,16)(5,21)(6,22)(7,29)(8,23)(9,30)(11,34)(12,20)(13,39)(14,35)(15,40)(17,42)(18,43)(19,45)(24,55)(25,52)(26,57)(27,53)(28,44)(31,58)(32,33)(36,50)(37,51)(38,49)(41,60)(46,64)(47,61)(48,63)(54,59)(56,62);(1,5,6,25,8,27,7,26)(2,11,12,36,14,24,13,37)(3,17,18,47,19,48,28,56)(4,21,22,52,23,53,29,57)(9,31,15,41,32,54,38,46)(10,34,20,50,35,55,39,51)(16,42,43,61,45,63,44,62)(30,58,40,60,33,59,49,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285536, #72285753, #72285960, …
maximal normal subgroup of	#72285536, #72285753, #72285960, …
maximal quotient of	C₂×#395175

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302371);