#245368

order	512
nilpotency class	3
isoclinism	isoclinic class of #245303
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18940	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152564);