#497458

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂²⋊C₄×D₈
prime radical	#62635	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

949 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)(3,16)(4,21,25,41)(5,6)(7,14)(8,32)(9,15)(10,18)(11,24,40,31)(12,13)(17,46,49,63)(19,44)(20,45)(22,38,54,58)(23,39,53,55)(26,29)(27,30)(28,35)(33,42)(34,59,60,56)(36,57)(37,51)(43,48,62,50)(47,52,64,61);(1,3)(2,10)(4,17)(5,16)(6,28)(7,19)(8,33)(9,20)(11,34)(12,18)(13,42)(14,35)(15,37)(21,52)(22,43)(23,56)(24,48)(25,49)(26,44)(27,45)(29,51)(30,57)(31,50)(32,36)(38,47)(39,46)(40,60)(41,61)(53,59)(54,62)(55,63)(58,64);(1,4)(2,11)(3,17)(5,22)(6,23)(7,31)(8,24)(9,25)(10,34)(12,38)(13,39)(14,41)(15,40)(16,43)(18,47)(19,50)(20,49)(21,32)(26,55)(27,54)(28,56)(29,58)(30,53)(33,48)(35,61)(36,52)(37,60)(42,46)(44,63)(45,62)(51,64)(57,59);(1,5)(2,12)(3,10)(4,22)(6,14)(7,26)(8,13)(9,27)(11,38)(15,29)(16,28)(17,34)(18,42)(19,35)(20,37)(21,53)(23,41)(24,39)(25,54)(30,32)(31,55)(33,36)(40,58)(43,56)(44,51)(45,57)(46,47)(48,52)(49,60)(50,61)(59,62)(63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314256
maximal normal subgroup of	#72314256

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404654);