#455118

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1438 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=ei,e^2=i,f^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,d^c=de,e^c=ei,f^c=fi,h^c=hi

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, 362314);