#496722

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#21973
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

443 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320359, #72320710, #72320755
maximal normal subgroup of	#72320359, #72320710, #72320755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403918);