#235515

order	512
nilpotency class	3
isoclinism	isoclinic class of #235504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37558	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1154 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=di,e^c=ei,f^c=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142711);