#448645

order	512
nilpotency class	3
isoclinism	isoclinic class of #448644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15914
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

916 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)(3,13,16,8)(4,18,22,37)(5,10,23,35)(6,25,28,11)(9,26,34,21)(14,41,19,50)(15,30,45,39)(17,31,46,40)(20,52,53,49)(24,38,27,36)(29,43,32,47)(33,55,60,54)(42,61,51,64)(44,58,48,57)(56,63,59,62);(1,3,6,17,7,16,28,46)(2,8,11,31,12,13,25,40)(4,19,21,47,22,14,26,43)(5,15,24,44,23,45,27,48)(9,32,18,41,34,29,37,50)(10,30,36,57,35,39,38,58)(20,51,54,63,53,42,55,62)(33,59,52,64,60,56,49,61);(1,4)(2,9)(3,14)(5,20)(6,26)(7,22)(8,29)(10,33)(11,37)(12,34)(13,32)(15,42)(16,19)(17,47)(18,25)(21,28)(23,53)(24,55)(27,54)(30,56)(31,41)(35,60)(36,49)(38,52)(39,59)(40,50)(43,46)(44,63)(45,51)(48,62)(57,64)(58,61);(1,5)(2,10)(3,15,16,45)(4,20)(6,27)(7,23)(8,30,13,39)(9,33)(11,38)(12,35)(14,42,19,51)(17,48,46,44)(18,49)(21,55)(22,53)(24,28)(25,36)(26,54)(29,56,32,59)(31,58,40,57)(34,60)(37,52)(41,61,50,64)(43,63,47,62)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319622, #72320000, #72320069
maximal normal subgroup of	#72319622, #72320000, #72320069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355841);