#424342

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1478 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)(5,6,10,16,12,17,11,34)(7,18,21,37,23,45,22,13)(8,24,27,41,29,40,28,14)(9,15,31,43,33,44,32,60)(19,47,25,38,50,35,49,46)(20,42,30,59,52,66,51,36)(26,53,56,64,58,63,57,39)(48,67,54,62,55,61,68,65);(5,7,12,23)(6,13,17,37)(8,25,29,49)(9,30,33,51)(10,22,11,21)(14,38,41,46)(15,42,44,66)(16,45,34,18)(19,28,50,27)(20,32,52,31)(24,47,40,35)(26,55,58,48)(36,60,59,43)(39,61,64,67)(53,62,63,65)(54,57,68,56);(5,8)(6,14,17,41)(7,19)(9,26)(10,27)(11,28)(12,29)(13,35,37,47)(15,39,44,64)(16,24,34,40)(18,46,45,38)(20,48)(21,25)(22,49)(23,50)(30,54)(31,56)(32,57)(33,58)(36,61,59,67)(42,65,66,62)(43,53,60,63)(51,68)(52,55);(2,4)(5,9,11,32,12,33,10,31)(6,15,34,60,17,44,16,43)(7,20,22,51,23,52,21,30)(8,26,28,57,29,58,27,56)(13,36,45,66,37,59,18,42)(14,39,40,63,41,64,24,53)(19,48,49,68,50,55,25,54)(35,61,38,62,47,67,46,65)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 331538);