#285385

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

576 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)(3,8)(4,16)(5,7)(6,11)(9,17)(10,20)(12,23)(13,28)(14,21)(15,24)(18,25)(19,26)(22,27)(29,31)(30,32)(33,34,38,43,39,44,37,42)(35,40,47,56,48,57,46,55)(36,49,51,59,52,60,53,41)(45,61,62,58,50,64,63,54);(2,7)(3,12)(5,20)(9,16)(10,11)(13,27)(14,15)(17,18)(21,23)(25,26)(28,31)(29,30)(33,35,38,47,39,48,37,46)(34,40,43,56,44,57,42,55)(36,50,51,63,52,45,53,62)(41,58,49,64,59,54,60,61);(1,3,6,15)(2,8,11,24)(4,13,19,29)(5,14,20,12)(7,21,10,23)(9,22,25,32)(16,28,26,31)(17,27,18,30)(33,36,39,52)(34,41,44,59)(35,45,48,50)(37,51,38,53)(40,54,57,58)(42,49,43,60)(46,62,47,63)(55,61,56,64);(1,4)(2,9)(3,13)(5,18)(6,19)(7,16)(8,22)(10,26)(11,25)(12,27)(14,30)(15,29)(17,20)(21,28)(23,31)(24,32)(33,37,39,38)(34,42,44,43)(35,46,48,47)(36,51,52,53)(40,55,57,56)(41,49,59,60)(45,62,50,63)(54,61,58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192581);