#461254

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 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)(7,18,23,46)(8,24)(9,15)(10,16)(11,34)(12,17)(13,21,37,22)(14,28)(19,35,25,55)(20,45,53,67)(26,40)(27,39)(29,43)(30,59)(31,41)(32,61)(33,42)(36,51,64,52)(38,57)(44,48,54,49)(47,62,56,68)(50,63,60,66)(58,65);(5,7,11,22,12,23,10,21)(6,13,16,18,17,37,34,46)(8,25,28,48,29,19,27,49)(9,20,32,52,33,53,31,51)(14,35,24,54,39,55,43,44)(15,36,41,45,42,64,61,67)(26,56,57,50,58,47,30,60)(38,62,40,63,59,68,65,66);(5,8,12,29)(6,14,17,39)(7,19,23,25)(9,30,33,57)(10,28,11,27)(13,35,37,55)(15,40,42,65)(16,43,34,24)(18,44,46,54)(20,50,53,60)(21,49,22,48)(26,32,58,31)(36,63,64,66)(38,61,59,41)(45,62,67,68)(47,52,56,51);(2,4)(5,9,11,32,12,33,10,31)(6,15,34,61,17,42,16,41)(7,20,22,52,23,53,21,51)(8,26,28,57,29,58,27,30)(13,36,46,67,37,64,18,45)(14,38,43,65,39,59,24,40)(19,47,49,60,25,56,48,50)(35,62,44,66,55,68,54,63)

Polycyclic presentation:

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

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, 368450);