#497452

order	512
nilpotency class	4
isoclinism	isoclinic class of #497451
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²⋊C₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3416	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

957 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)(3,14)(4,9,21,34)(5,23)(6,13)(7,10)(8,17)(11,24)(12,25)(15,40,43,55)(16,39)(18,29)(19,48)(20,49,52,58)(22,32,51,35)(26,36)(27,37)(28,54,56,42)(30,45)(31,46)(33,59,61,50)(38,47)(41,62,63,57)(44,64,53,60);(1,3,6,18)(2,8,13,31)(4,15,21,43)(5,16,26,45)(7,27,25,19)(9,28,34,56)(10,30,12,39)(11,37,38,48)(14,36,29,23)(17,47,46,24)(20,41,52,63)(22,53,51,44)(32,57,35,62)(33,60,61,64)(40,58,55,49)(42,59,54,50);(1,4)(2,9)(3,15)(5,20)(6,21)(7,22)(8,28)(10,32)(11,33)(12,35)(13,34)(14,40)(16,41)(17,42)(18,43)(19,44)(23,49)(24,50)(25,51)(26,52)(27,53)(29,55)(30,57)(31,56)(36,58)(37,60)(38,61)(39,62)(45,63)(46,54)(47,59)(48,64);(2,10)(3,16)(5,24)(7,25)(8,29)(9,32)(12,13)(14,31)(15,41)(17,19)(18,45)(20,50)(22,51)(23,36)(26,47)(27,46)(28,55)(30,37)(34,35)(39,48)(40,56)(42,44)(43,63)(49,58)(52,59)(53,54)(57,60)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319748, #72343955, #72343961
maximal normal subgroup of	#72319748, #72343955, #72343961

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404648);