#300518

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	#3561	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

816 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)(3,12)(4,16)(5,10)(6,20)(7,11)(8,14)(9,19)(13,30)(15,24)(17,23)(18,28)(21,25)(22,27)(26,31)(29,32)(33,34,40,46)(35,47,51,58)(36,43,54,61)(37,55,53,42)(38,45,39,44)(41,50,59,49)(48,56,64,57)(52,62,63,60);(1,3,5,14)(2,8,10,12)(4,17,18,32)(6,15,7,21)(9,26,27,30)(11,24,20,25)(13,22,31,19)(16,29,28,23)(33,35,38,49,40,51,39,50)(34,41,44,47,46,59,45,58)(36,52,53,64,54,63,37,48)(42,60,43,57,55,62,61,56);(1,4,6,19,5,18,7,22)(2,9,11,16,10,27,20,28)(3,13,15,32,14,31,21,17)(8,23,24,26,12,29,25,30)(33,36,38,53,40,54,39,37)(34,42,44,43,46,55,45,61)(35,48,50,63,51,64,49,52)(41,56,58,62,59,57,47,60);(2,10)(4,18)(8,12)(11,20)(13,31)(17,32)(19,22)(24,25)(33,37,39,54,40,53,38,36)(34,43,45,42,46,61,44,55)(35,48,50,63,51,64,49,52)(41,57,58,60,59,56,47,62)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=ei,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, 207714);