#497280

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×C₂²⋊C₈
prime radical	#3584	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225458, #72225465, #72225467
maximal normal subgroup of	#72225458, #72225465, #72225467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404476);