#300511

order	512
nilpotency class	3
isoclinism	isoclinic class of #300501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3559	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

808 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207707);