#245349

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	#18934	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^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, 152545);