#191278

order	512
nilpotency class	3
isoclinism	isoclinic class of #191275
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

463 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98474);