#456881

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1104 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)(5,6,10,16,12,18,11,17)(7,19,22,39,24,38,23,13)(8,25,29,43,30,42,28,14)(9,31,33,15,35,45,34,46)(20,36,26,57,50,56,49,40)(21,51,53,47,55,37,54,61)(27,58,59,44,60,41,32,62)(48,64,67,63,68,65,52,66);(1,3,4,2)(5,7,12,24)(6,13,18,39)(8,26)(9,21,35,55)(10,23,11,22)(14,40)(15,37,46,51)(16,38,17,19)(20,29)(25,56)(27,52)(28,50)(30,49)(31,61,45,47)(32,48)(33,54,34,53)(36,42)(41,65)(43,57)(44,66)(58,64)(59,68)(60,67)(62,63);(2,3)(5,8,10,28,12,30,11,29)(6,14,16,42,18,43,17,25)(7,20,22,49,24,50,23,26)(9,32,33,60,35,59,34,27)(13,36,19,40,39,56,38,57)(15,44,45,58,46,62,31,41)(21,52,53,68,55,67,54,48)(37,64,61,66,51,65,47,63);(5,9,12,35)(6,15,18,46)(7,21,24,55)(8,27,30,60)(10,34,11,33)(13,37,39,51)(14,41,43,58)(16,31,17,45)(19,47,38,61)(20,48,50,68)(22,54,23,53)(25,44,42,62)(26,52,49,67)(28,59,29,32)(36,63,56,66)(40,65,57,64)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=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, 364077);