#455218

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1241 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)(5,6,12,16,11,17,10,31)(7,18,33,37,22,46,21,13)(8,14,32,39,26,40,25,43)(9,27,34,42,30,57,29,15)(19,44,53,35,23,54,48,62)(20,49,60,67,52,36,51,45)(24,41,59,64,56,65,28,38)(47,61,58,63,55,68,50,66);(5,7,10,21,11,22,12,33)(6,13,16,18,17,37,31,46)(8,23,25,53,26,19,32,48)(9,20,29,51,30,52,34,60)(14,35,39,54,40,62,43,44)(15,36,27,45,42,49,57,67)(24,55,28,58,56,47,59,50)(38,61,41,63,64,68,65,66);(5,8,11,26)(6,14,17,40)(7,19,22,23)(9,28,30,59)(10,32,12,25)(13,35,37,62)(15,41,42,65)(16,43,31,39)(18,44,46,54)(20,50,52,58)(21,53,33,48)(24,34,56,29)(27,38,57,64)(36,63,49,66)(45,61,67,68)(47,60,55,51);(2,4)(5,9,12,34,11,30,10,29)(6,15,31,57,17,42,16,27)(7,20,33,60,22,52,21,51)(8,24,32,59,26,56,25,28)(13,36,46,67,37,49,18,45)(14,38,43,65,40,64,39,41)(19,47,53,58,23,55,48,50)(35,61,44,66,62,68,54,63)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,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, 362414);