#332021

order	512
nilpotency class	3
isoclinism	isoclinic class of #331999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3245	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

506 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239217);