#496303

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21788
prime radical	#3584	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

919 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224956, #72224958, #72224961, …
maximal normal subgroup of	#72224956, #72224958, #72224961, …
maximal quotient of	C₂×#496303

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403499);