#245322

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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

904 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=i,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, 152518);