#493072

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1120 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=i,e^2=hi,g^2=hi,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400268);