#285411

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

575 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)(3,13)(4,18)(5,12)(6,11)(7,16)(8,20)(9,19)(10,23)(14,29)(15,24)(17,26)(21,28)(22,25)(27,31)(30,32)(33,34,37,41,38,42,51,58)(35,39,44,53,45,54,62,64)(36,46,48,56,49,57,50,40)(43,59,60,55,47,63,61,52);(1,3,6,17)(2,7,11,25)(4,15,20,30)(5,22,23,16)(8,27,18,29)(9,24,28,32)(10,26,12,13)(14,19,31,21)(33,35,37,44,38,45,51,62)(34,39,41,53,42,54,58,64)(36,47,48,61,49,43,50,60)(40,55,46,63,56,52,57,59);(1,4)(2,8)(3,14)(5,19)(6,20)(7,24)(9,10)(11,18)(12,28)(13,27)(15,22)(16,30)(17,31)(21,23)(25,32)(26,29)(33,36,38,49)(34,40,42,56)(35,43,45,47)(37,50,51,48)(39,52,54,55)(41,57,58,46)(44,61,62,60)(53,63,64,59);(1,4)(2,9)(3,15)(5,21)(6,20)(7,24)(8,10)(11,28)(12,18)(13,29)(14,22)(16,31)(17,30)(19,23)(25,32)(26,27)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^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, 192607);