#497341

order	512
nilpotency class	4
isoclinism	isoclinic class of #497339
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#61083	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#577 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1103 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)(3,13)(4,19,23,38)(5,24)(6,27,29,11)(7,12)(8,16)(9,22)(10,25)(14,41,20,52)(15,33)(17,47,49,40)(18,31)(21,53,54,51)(26,37,58,57)(28,36)(30,61)(32,46,60,45)(34,64)(35,59)(39,44)(42,50,43,63)(48,62)(55,56);(1,3,7,18)(2,8,12,33)(4,20,23,14)(5,15,25,16)(6,17,29,49)(9,34,36,30)(10,31,24,13)(11,32,27,60)(19,50,38,63)(21,43,54,42)(22,48,28,44)(26,45,58,46)(35,39,56,62)(37,47,57,40)(41,53,52,51)(55,61,59,64);(1,4,7,23)(2,9,12,36)(3,14,18,20)(5,21,25,54)(6,28,29,22)(8,30,33,34)(10,35,24,56)(11,38,27,19)(13,39,31,62)(15,42,16,43)(17,48,49,44)(26,59,58,55)(32,50,60,63)(37,53,57,51)(40,52,47,41)(45,61,46,64);(3,15)(5,25)(8,31)(10,24)(13,33)(14,42)(16,18)(17,45)(20,43)(21,54)(26,58)(30,62)(32,47)(34,39)(35,56)(37,57)(40,60)(41,50)(44,64)(46,49)(48,61)(51,53)(52,63)(55,59)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312286, #72312369, #72312372, …
maximal normal subgroup of	#72312286, #72312369, #72312372, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404537);