#423640

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
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

1503 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330836);