#503280

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3416	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1055 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=ehi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 410476);