#245358

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	#3548	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

929 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:

(1,2,7,13)(3,14,18,29)(4,20,5,10)(6,24,8,12)(9,19,26,17)(11,22,28,23)(15,27,31,30)(16,25,21,32)(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);(1,3,7,18)(2,9,13,26)(4,21,5,16)(6,19,8,17)(10,27,20,30)(11,25,28,32)(12,29,24,14)(15,23,31,22)(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);(1,4,7,5)(2,10,13,20)(3,15,18,31)(6,22,8,23)(9,25,26,32)(11,12,28,24)(14,30,29,27)(16,17,21,19)(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);(1,5,7,4)(2,11,13,28)(3,16,18,21)(6,22,8,23)(9,25,26,32)(10,12,20,24)(14,27,29,30)(15,17,31,19)(37,51)(41,49)(44,63)(46,57)(48,50)(53,61)(58,64)(60,62)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=h,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, 152554);