#417689

order	512
nilpotency class	3
isoclinism	isoclinic class of #417688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15257
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2325 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,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	#72286176, #72286213, #72286267
maximal normal subgroup of	#72286176, #72286213, #72286267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324885);