#332020

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^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, 239216);