#434072

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1131 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)(5,6,12,16,11,18,10,17)(7,19,23,37,24,45,22,13)(8,25,27,41,28,40,34,14)(9,29,33,44,32,15,31,43)(20,38,49,46,51,35,50,63)(21,42,52,61,53,47,30,36)(26,54,56,39,57,65,58,59)(48,62,55,66,68,60,67,64);(5,7,10,22,11,24,12,23)(6,13,16,19,18,37,17,45)(8,20,27,49,28,51,34,50)(9,30,31,53,32,52,33,21)(14,38,40,63,41,35,25,46)(15,42,43,61,29,47,44,36)(26,55,56,68,57,67,58,48)(39,64,54,60,59,66,65,62);(5,8,11,28)(6,14)(7,20)(9,26,32,57)(10,27,12,34)(13,35,37,38)(15,39)(16,40)(17,25)(18,41)(19,46,45,63)(21,48)(22,49)(23,50)(24,51)(29,59)(30,55)(31,56,33,58)(36,60,61,62)(42,66,47,64)(43,54)(44,65)(52,67)(53,68);(2,4)(5,9,11,32)(6,15,18,29)(7,21,24,53)(8,26,28,57)(10,33,12,31)(13,36,37,61)(14,39,41,59)(16,44,17,43)(19,47,45,42)(20,48,51,68)(22,52,23,30)(25,54,40,65)(27,58,34,56)(35,60,38,62)(46,64,63,66)(49,67,50,55)

Polycyclic presentation:

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

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, 341268);