#493818

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

940 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)(5,6,10,16,12,17,11,32)(7,18,23,44,21,42,24,52)(8,14,27,39,25,37,28,56)(9,29,30,40,31,57,19,15)(13,33,36,51,35,45,43,22)(20,41,48,63,50,64,49,67)(26,54,55,61,53,68,46,38)(34,58,59,66,60,65,62,47);(5,7,19,45,11,24,31,51,12,21,30,33,10,23,9,22)(6,13,29,44,16,36,40,42,17,35,57,52,32,43,15,18)(8,20,46,65,28,49,53,66,25,50,55,58,27,48,26,47)(14,34,54,63,39,59,61,64,37,60,68,67,56,62,38,41);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,25)(13,34)(15,38)(16,39)(17,37)(18,41)(19,46)(21,50)(22,47)(23,48)(24,49)(29,54)(30,55)(31,53)(32,56)(33,58)(35,60)(36,59)(40,61)(42,64)(43,62)(44,63)(45,65)(51,66)(52,67)(57,68);(7,21)(8,25)(13,35)(14,37)(18,42)(22,51)(23,24)(26,53)(27,28)(33,45)(36,43)(38,61)(39,56)(44,52)(46,55)(54,68)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401014);