#245373

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

915 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)(4,7)(5,20)(6,19)(8,11)(9,14)(10,21)(13,24)(15,26)(16,17)(18,23)(22,25)(27,28)(29,31)(30,32)(33,34,38,42)(35,39,45,54)(36,46,47,57)(37,49,51,41)(40,50,56,48)(43,58,59,64)(44,61,63,53)(52,62,55,60);(1,3,7,15)(2,9,4,17)(5,13,22,29)(6,16,8,14)(10,24,18,31)(11,26,19,12)(20,27,25,32)(21,30,23,28)(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);(1,4,7,2)(3,12,15,26)(5,18,22,10)(6,19,8,11)(9,14,17,16)(13,27,29,32)(20,21,25,23)(24,30,31,28)(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)(2,10)(3,13)(4,18)(6,23)(7,22)(8,21)(9,24)(11,25)(12,27)(14,30)(15,29)(16,28)(17,31)(19,20)(26,32)(37,51)(41,49)(44,63)(46,57)(48,50)(53,61)(58,64)(60,62)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^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, 152569);