#413472

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6,11,17)(7,19,23,37)(8,14,28,41)(9,29,31,42)(10,18,12,16)(13,24,36,22)(15,30,43,32)(20,38,48,59)(21,35,50,61)(25,34,47,60)(26,53,55,64)(27,44,33,40)(39,57,63,54)(45,49,56,51)(46,58,66,68)(52,67,65,62);(5,7,12,24,11,23,10,22)(6,13,18,37,17,36,16,19)(8,25,27,20,28,47,33,48)(9,21,32,51,31,50,30,49)(14,38,40,34,41,59,44,60)(15,35,29,56,43,61,42,45)(26,52,54,46,55,65,57,66)(39,62,53,58,63,67,64,68);(5,8,12,27,11,28,10,33)(6,14,18,40,17,41,16,44)(7,20,24,47,23,48,22,25)(9,26,32,54,31,55,30,57)(13,34,37,59,36,60,19,38)(15,39,29,53,43,63,42,64)(21,46,51,65,50,66,49,52)(35,58,56,67,61,68,45,62);(1,2,3,4)(5,9,10,30,11,31,12,32)(6,15,16,42,17,43,18,29)(7,21,22,49,23,50,24,51)(8,26,33,57,28,55,27,54)(13,35,19,45,36,61,37,56)(14,39,44,64,41,63,40,53)(20,46,25,52,48,66,47,65)(34,58,38,62,60,68,59,67)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=ef,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320668);