#285430

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	C₂×#41	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

566 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)(4,17)(5,7)(6,11)(8,16)(9,18)(10,21)(13,22)(14,29)(15,24)(19,26)(20,25)(23,31)(27,28)(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)(3,13)(5,21)(9,17)(10,11)(14,28)(15,16)(18,20)(22,24)(25,26)(29,32)(30,31)(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)(2,8)(4,14)(5,15)(6,16)(7,22)(9,23)(10,24)(11,12)(13,21)(17,32)(18,28)(19,30)(20,31)(25,27)(26,29)(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);(1,4,6,19)(2,9,11,25)(3,14,16,30)(5,20,21,18)(7,17,10,26)(8,23,12,27)(13,28,15,31)(22,32,24,29)(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=fh,b^2=fh,c^2=i,d^2=hi,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, 192626);