#420422

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1634 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ei,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,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, 327618);