#332006

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	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

502 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,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, 239202);