#428078

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,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, 335274);