#497406

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
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	D₄×#102
prime radical	#61105	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1035 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404602);